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for proper , locally flat embeddings in @xmath1 , it is well - known that :    1 . [ rlt ] a ray (= copy of @xmath2 ) knots if and only if @xmath3 . [ hlt ] a hyperplane (= copy of @xmath4 ) knots if and only if @xmath3 .    both facts hold in the smooth , piecewise linear , and topological categories  @xcite . fox and artin discovered the first knotted ray  @xcite . the boundary of a closed regular neighborhood of any knotted ray is a knotted hyperplane . for @xmath5 , fact [ hlt ] is the cantrell - stallings hyperplane unknotting theorem , an enhancement of the famous schoenflies theorem of mazur and brown @xcite , . embeddings in @xmath1 , @xmath6 , of at most countably many rays or hyperplanes were recently classified by king , siebenmann , and the first author  @xcite . in @xmath0 , no classification is known or even conjectured . + a ray or multiray @xmath7 is _ unknotted _ if and only if an automorphism of @xmath0 carries @xmath8 to a union of radial rays . unknotted multirays with the same number of components are ambient isotopic  ( * ? ? ? * lemma  4.1 ) . + a rich collection of knotted rays may be obtained from wild arcs . let @xmath9 be an arc with one wild (= non - locally flat ) point @xmath10 . consider @xmath11 in @xmath12 . if @xmath10 is an endpoint of @xmath13 , then @xmath8 is a knotted ray . if @xmath10 is an interior point of @xmath13 , then @xmath8 is a knotted , two component multiray . hence , in @xmath0 :    1 .   there exist infinitely many knot types of a ray  @xcite . 2 .   there exist uncountably many knot types of a ray  @xcite . 3 .   there exist uncountably many knot types of two component multirays with unknotted components  @xcite . a three component multiray @xmath7 will be called _ borromean rays _ provided @xmath8 is knotted , but any two components of @xmath8 form an unknotted multiray . debrunner and fox constructed an example equivalent to borromean rays  @xcite . earlier , doyle attempted a construction  @xcite , but his argument contained a gap  @xcite . we prove that there exist uncountably many knot types of borromean rays . the following is an overview . + consider the four blocks in figure  [ four_blocks ] . , @xmath14 , @xmath15 , and @xmath16 . each block is a three component tangle in a thickened @xmath17-sphere . the set of these four blocks is denoted by @xmath18 . ] the block @xmath19 consists of a three component tangle @xmath20 in a thickened @xmath17-sphere @xmath21}$ ] . any two components of @xmath20 can be straightened by an ambient isotopy of @xmath21}$ ] relative to boundary . however , no diffeomorphism of @xmath21}$ ] sends @xmath20 to a radial tangle ( corollary  [ anottrivial ] ) . the blocks @xmath14 , @xmath15 , and @xmath16 are reflections of @xmath19 . let @xmath22 be the set of these four blocks . let @xmath23 , @xmath24 , be a sequence of blocks in @xmath18 . the _ infinite concatenation _ @xmath25 is obtained by gluing the inner boundary sphere of @xmath26 to the boundary of a @xmath27-disk , and gluing the inner boundary sphere of @xmath28 to the outer boundary sphere of @xmath23 for each @xmath24 . this yields the pair : @xmath29 where @xmath30 is a three component multiray . each such @xmath31 forms borromean rays ( corollary  [ borr_blocks_yield_borr_rays ] ) . let @xmath32 be the borromean rays determined by another such sequence @xmath33 , @xmath24 . we prove that if @xmath34 is a diffeomorphism of pairs , then there is an isotopy of @xmath35 to a diffeomorphism @xmath36 and an integer @xmath37 such that : @xmath38 hence , the existence of @xmath35 boils down to : ( i ) the tails of the sequences @xmath23 and @xmath33 , and ( ii ) possible diffeomorphisms between individual blocks in @xmath18 . the latter are studied in section  [ s : diffeo_blocks ] . our main result , theorem  [ borr_rays_thm ] , gives necessary and sufficient conditions for two such sequences to yield equivalent borromean rays . care is taken to account for orientation . as an application , we give necessary and sufficient conditions for our borromean rays to be achiral ( corollary  [ chiral_cor ] ) . while most turn out to be chiral , we give a countably infinite family of pairwise inequivalent , achiral borromean rays . + the notion of an _ irreducible block _ plays a central role . a block @xmath39 is _ irreducible _ provided : if @xmath39 is diffeomorphic to a concatenation @xmath40 , then @xmath26 or @xmath41 is diffeomorphic to a _ trivial block _ (= block with a radial tangle ) . trivial blocks are irreducible ( proposition  [ sigma_en ] ) . we use this fact to prove that each @xmath31 in   forms borromean rays . the block @xmath19 is also irreducible ( theorem  [ a_irred ] ) , although the proof is more technical . thus , blocks in @xmath18 are irreducible . this fact is used to improve diffeomorphisms as in  . + we are unaware of a general method for detecting irreducibility . for instance , let @xmath26 and @xmath41 be blocks containing @xmath37 component tangles @xmath42 and @xmath43 respectively . let @xmath31 be the tangle in the concatenation @xmath40 . let @xmath44 , @xmath45 , and @xmath46 be the fundamental groups of @xmath47 , @xmath48 , and @xmath49 respectively . let @xmath50 be the @xmath17-sphere where @xmath26 and @xmath41 meet in @xmath40 . then , @xmath51 is an @xmath37-punctured sphere and @xmath52 is free of rank @xmath53 . using dehn s lemma and the loop theorem  @xcite , one may show that the inclusions @xmath54 induce injective homomorphisms on fundamental groups . by van kampen s theorem , @xmath55 is the free product of @xmath44 and @xmath45 amalgamated over @xmath56 ( see  @xcite ) . by grushko s theorem  @xcite , the rank of the free product @xmath57 equals @xmath58 . thus , one might hope that @xmath59 . however , no such relation holds in general for free products with amalgamation  @xcite . still , rank behaves better when the amalgamating subgroup is malnormal in each factor  @xcite , @xcite . for knot groups , malnormality of the peripheral subgroups was studied recently by weidmann  @xcite and de la harpe and weber  @xcite . it is unclear to us whether @xmath60 is malnormal in @xmath44 for an arbitrary block @xmath26 . it would be interesting to find block invariants sensitive to irreducibility . + we discovered the block @xmath19 as follows . consider a three component multiray @xmath30 with the property :    1 . [ two_comps_std ] any two components of @xmath31 form an unknotted multiray . let @xmath61 , @xmath62 , denote the components of @xmath31 . property  [ two_comps_std ] implies that for each pair @xmath61 and @xmath63 of components of @xmath31 there is a _ (= properly embedded copy of @xmath65\times[0,\infty)$ ] ) whose _ stringers _ (= @xmath66 and @xmath67 ) equal @xmath61 and @xmath63 . the interior of @xmath68 probably intersects the third component of @xmath31 ( if not , then @xmath31 is unknotted ) . using a small regular neighborhood of @xmath31 , one may twist these strips about their stringers so that they patch together to form a general position immersion @xmath69 . let @xmath70 , @xmath62 , be equally spaced points in @xmath71 . the immersion @xmath35 sends the radial ray @xmath72 to @xmath61 for each @xmath62 and is an embedding on each of the three closed sectors between such radial rays . in an attempt to unknot @xmath31 , one may try to eliminate multiple points of @xmath35 . certain types of multiple points can be eliminated . however , difficulties arise from essential circles of double points . figure  [ immersion ] displays the relevant , compact part of a simple configuration where two essential circles of double points are identified under @xmath35 .     from the compact , @xmath17-dimensional annulus @xmath73 into the thickened sphere @xmath21}$ ] . the image of @xmath35 is a torus sitting atop an annulus and containing the tangle @xmath20 . ]    here , the domain of @xmath35 is the compact , @xmath17-dimensional annulus @xmath73 containing three radial arcs . the image of these three radial arcs is the tangle @xmath20 in the block @xmath19 . having found @xmath19 , a fundamental group calculation shows that @xmath19 is not a trivial block ( see section  [ equiv_rel_blocks ] ) . our proof that @xmath19 is irreducible ( see section  [ ab_irred ] ) makes essential use of the immersion @xmath35 in figure  [ immersion ] . + after we discovered the block @xmath19 , we found debrunner and fox s _ mildly wild @xmath27-frame _ @xcite , @xcite and doyle s attempted example  @xcite . we pause to make some observations on these two examples .    1 . debrunner and fox s mildly wild @xmath27-frame @xmath74 is compact , periodic , and contains one wild point @xmath75 . in our notation , their building block is the concatenation @xmath76 . put @xmath77 in @xmath78 . we define the _ debrunner - fox borromean rays _ to be @xmath79 in @xmath80 . in our notation : @xmath81 by corollary  [ chiral_cor ] below , @xmath82 is achiral . debrunner and fox s proof that @xmath77 is wild hinges on showing a certain group is not finitely generated . their approach yields a mildly wild @xmath37-frame for each @xmath83 . on the other hand , it is not clear how one can use it to distinguish between two wild @xmath27-frames . in section  [ irred_blocks ] , we use irreducibility of trivial blocks to prove our multirays ( including @xmath82 ) are knotted . then , we use irreducibility of blocks in @xmath18 to distinguish between multirays . bing showed that doyle s @xmath27-frame is standard  @xcite , though bing s argument is not indicated . lemma  [ multiray_straightening ] below is useful for recognizing unknotted multirays and applies to doyle s @xmath27-frame . this paper is organized as follows . section  [ definitions ] presents conventions and notation , introduces blocks ( including several examples ) , and proves some basic properties concerning blocks . section  [ s : diffeo_blocks ] studies diffeomorphisms between individual blocks in @xmath18 . section  [ irred_blocks ] introduces irreducible blocks , proves trivial blocks are irreducible , deduces some corollaries , and constructs infinitely many irreducible blocks containing two component tangles . section  [ ball_arc_pairs ] identifies some unknotted ball - arc pairs in blocks . section  [ ab_irred ] proves that blocks in @xmath18 are irreducible . section  [ improve_spheres ] simplifies certain spheres in concatenations of borromean blocks and deduces two useful corollaries . section  [ brbh ] classifies borromean rays arising from sequences of blocks in @xmath18 and then uses regular neighborhoods of multirays to obtain results on knotted multiple hyperplane embeddings . in particular , we prove that there exist uncountably many pairwise inequivalent so - called _ borromean hyperplanes _ in @xmath0 . we work in the smooth (= @xmath84 ) category . throughout , @xmath85 denotes diffeomorphism of manifolds or manifold pairs . a map is * proper * provided the inverse image of each compact set is compact . all isotopies will be smooth and proper . a submanifold @xmath86 is * neat * provided @xmath87 and this intersection is transverse  @xcite , @xcite . + a * ray * is a proper embedding of @xmath2 . a * multiray * is a proper embedding of @xmath88 where @xmath89 is a finite or countably infinite discrete space . indeed , each embedded submanifold of @xmath1 contains at most countably many components since @xmath1 is a separable metric space . a ray in @xmath1 is * radial * provided either it is straight and emanates from the origin , or it is contained in such a ray . in particular , a radial ray can meet the origin only at its endpoint . a collection of intervals embedded in @xmath1 is * radial * provided each component lies in a radial ray . + the standard euclidean norm on @xmath1 is @xmath90 . on @xmath0 , the euclidean norm function will be denoted : @xmath91 ^ -{\eta }       &        \mathbb{r}\\      x                \ar@{|-{>}}[r ]   &        \left\|x\right\| } \end{split}\ ] ] all lengths come from the standard euclidean metric . the unit @xmath37-disk @xmath92 consists of all points @xmath93 such that @xmath94 . the unit @xmath95-sphere is @xmath96 . the sphere of radius @xmath97 in @xmath1 about @xmath98 is denoted @xmath99}$ ] and is called a * level sphere*. in particular , @xmath100}$ ] . let @xmath101}$ ] , where @xmath102 , denote the thickened sphere of points @xmath93 such that @xmath103 . in particular , @xmath104}$ ] equals the disjoint union of the spheres @xmath105}$ ] and @xmath106}$ ] . let @xmath107 denote the half - infinite annulus of points @xmath93 such that @xmath108 . let @xmath109}$ ] , where @xmath102 , be a thickened sphere in @xmath0 . a * tangle * @xmath31 is an embedding of the disjoint union of @xmath110 copies of @xmath65 $ ] as a neat submanifold of @xmath109}$ ] . if @xmath111 is a component of @xmath31 , then the initial point @xmath112 of @xmath111 must lie in @xmath113}$ ] and the terminal point must equal @xmath114}$ ] . so , @xmath111 stretches between the two boundary @xmath17-spheres of @xmath115}$ ] , and its initial and terminal points lie on a radial ray . + a * block * is a pair @xmath116},\tau)$ ] where @xmath31 is a tangle . each block is oriented : @xmath115}$ ] inherits its orientation from the standard one on @xmath0 , and each component of @xmath31 is oriented to point out from the inner boundary @xmath17-sphere . a * diffeomorphism * of blocks is any diffeomorphism of the corresponding pairs of spaces , not necessarily orientation or boundary preserving in any sense . + a * trivial block * is any pair @xmath117 consisting of an @xmath37 component , radial tangle in a thickened sphere ( see figure  [ trivial_block ] ) . +     containing a three component , radial tangle . ]    by our convention , in every displayed block , the positive @xmath118-axis points horizontally to the right , the positive @xmath119-axis points vertically up , and the positive @xmath120-axis points out of the page towards the reader . + given any block @xmath121},\tau)$ ] , define two blocks :    1 . @xmath122},\overline{\tau})$ ] is the reflection of @xmath39 across the @xmath123-plane . @xmath124},\tau^{\ast})$ ] is the inversion of @xmath39 across @xmath125}$ ] . if @xmath126},\tau)$ ] , then inversion is @xmath127 . components of @xmath128 are still oriented out from the inner boundary @xmath17-sphere . evidently , the bar and star operations commute and are involutions : @xmath129    figure  [ four_blocks ] above introduced four blocks important for our purposes . let : @xmath130 be the set of these four blocks . note that @xmath18 is closed under the bar and star operations . by construction , blocks in @xmath18 are pairwise diffeomorphic . they are pairwise distinct , though , up to finer equivalence relations , as explained below . the coarsest equivalence relation on blocks we consider is that of _ diffeomorphism _ ( defined above ) . finer diffeomorphism relations , involving orientation and/or boundary preservation , arise in section  [ s : diffeo_blocks ] . on blocks with the same underlying thickened spheres , the finest relation we consider ( short of equality ) is that of * ambient isotopy relative to boundary * , meaning ambient isotopy of tangles fixing both boundary @xmath17-spheres pointwise at all times . [ one_comp ] any block @xmath131},\tau\right)}$ ] , where @xmath31 has one component , is ambient isotopic relative to boundary to @xmath132 . this fact is well - known ( e.g. , it s an exercise in rolfsen @xcite ) . we are not aware of a published proof , so we sketch one . all isotopies are ambient and relative to boundary . by a preliminary isotopy ( left to the reader ) , we assume @xmath39 appears as in figure  [ knot_block_e1 ] where @xmath133 is a crossing diagram in general position .     with a one component tangle @xmath31 . ]    it suffices to prove that crossings of @xmath133 may be switched by isotopy , since then we may arrange that @xmath133 has monotonic @xmath120-coordinate and the result follows . so , consider a crossing @xmath134 of @xmath133 with over arc @xmath135 and under arc @xmath136 . push @xmath135 and @xmath136 sufficiently close together in the @xmath120-direction . let @xmath10 and @xmath137 be the midpoints of @xmath135 and @xmath136 respectively , where @xmath10 lies directly above @xmath137 . let @xmath138 $ ] denote the subarc of @xmath31 where @xmath139 and @xmath140 . let @xmath141 $ ] if @xmath142 $ ] , and let @xmath143 $ ] if @xmath144 $ ] . in other words , @xmath145 is the unique subarc of @xmath31 originating on @xmath146 and terminating at the first point , @xmath10 or @xmath137 , encountered by @xmath145 . assume @xmath141 $ ] ( otherwise , flip the picture over ) . isotop @xmath137 close to @xmath146 by following just underneath @xmath145 and stretching @xmath136 . then , loop @xmath136 under @xmath146 and isotop @xmath137 back ( again using @xmath145 as a guide ) to lie above @xmath10 . the crossing @xmath134 has been switched , completing the proof . [ knots_tied ] example  [ one_comp ] has the following possibly surprising corollary , which appears to be due to wilder  @xcite . if the ray @xmath7 is obtained by tying successive knots in a radial ray ( see figure  [ knots_ray ] ) ,     obtained by tying successive knots in a radial ray . ]    then @xmath8 is ambient isotopic to a radial ray . proof : let @xmath147}$ ] for each @xmath148 . simultaneously apply the straightening process from example  [ one_comp ] to each @xmath149}$ ] . @xmath150    [ knot_blocks ] let @xmath151 be a knot . let @xmath152 be a @xmath27-disk such that : ( i ) @xmath153 is a neatly embedded arc in @xmath77 , and ( ii ) @xmath154 is an unknotted ball - arc pair . let @xmath155 be a small , round @xmath27-disk meeting one tangle component in an arc . the * knot block * @xmath156 is obtained from @xmath157 by replacing @xmath158 with @xmath159 as in figure  [ knot_block ] .     where @xmath133 is a diagram yielding @xmath160 . ]    in general , @xmath156 is well - defined up to diffeomorphism . if @xmath160 itself is oriented , then one could define @xmath156 more carefully . let @xmath31 be the tangle in a knot block @xmath156 . evidently , deleting the boundary from @xmath161 yields @xmath162 . in particular , @xmath163 implies @xmath164 . so , knots with nonisomorphic groups ( e.g. , torus knots  @xcite ) yield nondiffeomorphic knot blocks . finally , let @xmath131},\tau\right)}$ ] be a block where @xmath31 has two components , @xmath42 and @xmath43 . then , @xmath39 is ambient isotopic relative to boundary to some knot block . to see this , straighten @xmath43 using the process in example  [ one_comp ] . then push @xmath42 away from @xmath43 by integrating a suitable vector field tangent to level spheres . [ block_d ] consider the block @xmath77 in figure  [ one_twist ] .    . ]    the thickened sphere underlying @xmath77 is @xmath21}$ ] , and @xmath77 is obtained from @xmath165 by fixing the inner boundary @xmath17-sphere pointwise and rigidly rotating the outer boundary sphere one revolution about the @xmath118-axis . in particular , @xmath166 , in fact by an orientation preserving diffeomorphism that is pointwise the identity on both boundary @xmath17-spheres . on the other hand , @xmath165 and @xmath77 are not ambient isotopic relative to boundary , as proved by newman  @xcite and fadell  @xcite ( see also  @xcite and  @xcite ) . if @xmath77 had been obtained from @xmath165 by two complete twists , rather than just one , then @xmath77 would have been ambient isotopic relative to boundary to @xmath165 by dirac s belt trick . a diffeomorphism between thickened spheres is : ( i ) * radial * provided it sends radial arcs to radial arcs , and ( ii ) * level * provided it sends level @xmath17-spheres to level @xmath17-spheres . the next lemma says that there is essentially just one trivial block @xmath117 for each @xmath167 . [ en_unique ] let @xmath168},r)$ ] and @xmath169},r')$ ] be trivial blocks . then , there is a radial , level , orientation preserving diffeomorphism @xmath170 sending @xmath171}$ ] to @xmath172}$ ] for @xmath173 . if @xmath174}={\left[t'_1,t'_2\right]}$ ] , then there is an ambient isotopy @xmath175 , @xmath176 , of @xmath115}$ ] such that :    1 . [ eh0id ] @xmath177 and @xmath178 . [ h_tradial ] @xmath175 is a radial , level diffeomorphism for all @xmath176 . let @xmath179}\to{\left[t'_1,t'_2\right]}$ ] be the unique affine , orientation preserving diffeomorphism . then , @xmath180 is a radial , level diffeomorphism @xmath115}\to s^2{\left[t'_1,t'_2\right]}$ ] . so , it suffices to consider the case @xmath174}={\left[t'_1,t'_2\right]}=[1,2]$ ] . let @xmath181 , @xmath182 , denote the initial points of the components @xmath183 of @xmath8 . define @xmath184 similarly for @xmath185 . let @xmath186 be a smooth , simple path from @xmath187 to @xmath188 in @xmath146 . let @xmath189 be a smooth regular neighborhood of @xmath186 in @xmath146 . there is an ambient isotopy of @xmath146 , with support in @xmath189 , carrying @xmath187 to @xmath188 . for instance , begin with a suitable nonzero tangent vector field to @xmath186 , extend to a vector field @xmath190 on @xmath146 that vanishes outside of @xmath189 , and then integrate @xmath190 ( cf . extend this isotopy radially to get an ambient isotopy of @xmath21}$ ] carrying @xmath191 to @xmath192 . any component @xmath183 , @xmath193 , that moved during this isotopy is still radial and is still denoted @xmath183 . repeat this procedure , while choosing @xmath194 disjoint from @xmath195 .    to distinguish @xmath19 from @xmath165 up to diffeomorphism , it suffices to distinguish the fundamental groups of their tangle complements up to isomorphism . presentations of such groups are obtained using wirtinger s algorithm . consider the diagram of @xmath19 in figure  [ wirt_a ] .     with oriented and labeled arcs . ]    as usual , labels of arcs correspond to generators of @xmath196 . the basepoint is above the page . the based loop representing a generator @xmath197 first penetrates the plane of the page at a point just to the right of the oriented arc labeled @xmath197 , and has linking number @xmath198 with this oriented arc . a presentation of @xmath199 is : @xmath200 each crossing of @xmath20 contributes a relation . the last relation is evident topologically . vertex relation _ for the fundamental group of the complement of the graph obtained by crushing the inner boundary @xmath17-sphere to a point , , . the outer vertex relation , @xmath201 , is redundant . + for the trivial tangle , @xmath202 is free of rank @xmath17 . to distinguish @xmath196 from @xmath203 , we count their _ classes _ of homomorphisms into small symmetric groups @xmath204 using the computer algebra system magma ( a finite problem ) . two homomorphisms @xmath205 are considered _ equivalent _ provided there exists @xmath206 such that @xmath207 for all @xmath208 . table  [ magma_data_1 ] collects this data . .numbers of classes of homomorphisms into @xmath204 . [ cols="^,^,^",options="header " , ]     [ non_exist_types ] let @xmath209 be blocks in @xmath18 . then , the type @xmath210 is not in the image of @xmath211 . suppose otherwise . then , there is a diffeomorphism @xmath212 of type @xmath210 . + let @xmath213 denote the @xmath17-sphere @xmath146 with three marked points @xmath214 . the mapping class group @xmath215 is the group of orientation preserving diffeomorphisms of @xmath146 that send @xmath216 , modulo isotopies of @xmath146 fixing @xmath187 , @xmath217 , and @xmath218 at all times  @xcite . recall that the natural map @xmath219^{\cong } & { \textnormal{sym}}(3)}\ ] ] which sends an element of @xmath215 to its action on the three marked points , is an isomorphism  @xcite . + by hypothesis , @xmath220 preserves boundary @xmath17-spheres componentwise and preserves orientation of @xmath21}$ ] . so , @xmath221 is an orientation preserving diffeomorphism . as the tangle permutation of @xmath220 is the identity ,   permits us to assume @xmath220 is the identity on inner boundary @xmath17-spheres . pasting together the diffeomorphisms @xmath222 and @xmath212 yields a diffeomorphism @xmath223 . this contradicts the last row of table  [ magma_data_2 ] since @xmath209 . consider two types @xmath224 such that @xmath225 is defined . as the image of @xmath211 is a subgroupoid of @xmath226 , if any two of @xmath135 , @xmath136 , or @xmath225 lie in the image of @xmath211 , then the third does as well . therefore , if @xmath135 is any type forbidden by lemma  [ non_exist_types ] , @xmath136 is any of the already realized @xmath227 types in the image of @xmath211 , and @xmath225 is defined in @xmath226 , then @xmath225 is not in the image of @xmath211 . a tedious , but completely straightforward calculation ( facilitated by magma ) , shows that this yields @xmath228 types not in the image of @xmath211 . this completes our proof of table  [ hom_block ] , and yields the following . [ blocks_inequiv_isotopy ] the blocks @xmath19 , @xmath14 , @xmath15 , and @xmath16 are pairwise distinct up to ambient isotopy relative to boundary . this section introduces the notion of an _ irreducible block_. such blocks play a central role in our construction of borromean rays . first , recall the following standard definitions and accompanying lemma . + a @xmath17-sphere @xmath50 in a thickened sphere is * essential * provided it does not bound a @xmath27-disk in the thickened sphere . otherwise , @xmath50 is * inessential*. let @xmath13 be an arc transverse to a @xmath17-manifold @xmath50 in a @xmath27-manifold . then , @xmath229 denotes the number of points in @xmath230 ( ignoring any orientations ) . the * mod @xmath17 intersection number * of @xmath13 and @xmath50 , denoted @xmath231 , is @xmath232 . [ ess_sphere ] let @xmath13 be a neatly embedded arc in @xmath115}$ ] . assume @xmath13 has one boundary point in @xmath113}$ ] and the other in @xmath233}$ ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath115}$ ] and transverse to @xmath13 . the following are equivalent : ( i ) @xmath50 is essential in @xmath115}$ ] , ( ii ) @xmath234 , and ( iii ) there is a neighborhood @xmath235 of @xmath236}$ ] in @xmath115}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath115}$ ] such that :    1 .   @xmath177 . @xmath238 for all @xmath176 . 3 .   @xmath239 is a level @xmath17-sphere in @xmath240}}$ ] . assume , without loss of generality , that @xmath174}=[1,2]$ ] . let @xmath77 denote , the @xmath27-disk of radius @xmath17 . by the @xmath27-dimensional schoenflies theorem ( * ? iii ) , @xcite , ( * ? ? ? 1.1 ) , @xmath50 bounds a unique @xmath27-disk @xmath241 . let @xmath242 . so , @xmath243 and @xmath244.case 1 . then , @xmath246 lies in the interior of @xmath247 $ ] . so , @xmath50 is inessential and clearly @xmath248.case 2 . @xmath249 then , @xmath50 can not bound a @xmath27-disk in @xmath247 $ ] and , hence , is essential in @xmath247 $ ] . the arc @xmath13 has one boundary point in @xmath250 and one outside @xmath246 , so @xmath251 . let @xmath252 be a level @xmath17-sphere between @xmath50 and @xmath253}$ ] . let @xmath158 be the @xmath27-disk in @xmath254 with boundary @xmath252 . let @xmath255 be the compact region in @xmath254 with boundary @xmath256 . by uniqueness of disk embeddings  @xcite , there is an ambient isotopy of @xmath158 carrying @xmath246 to a round @xmath27-disk . hence , there is a diffeomorphism @xmath257 $ ] sending @xmath50 to @xmath258 . construct a vector field @xmath190 on @xmath247 $ ] as follows . on @xmath259 , @xmath190 is the pushforward by @xmath260 of the constant vector field @xmath261 on @xmath262 $ ] . extend @xmath190 to the rest of @xmath21}$ ] , making it @xmath98 outside a small neighborhood of @xmath259 . the isotopy generated by @xmath190 is the desired @xmath175 . next , we give two definitions of * irreducible block * and then we prove they are equivalent . let @xmath121},\tau)$ ] be a block where @xmath31 has @xmath110 components . [ irred_def_1 ] the block @xmath39 is * irreducible * provided : if @xmath39 is diffeomorphic to a concatenation of blocks @xmath40 , then @xmath26 or @xmath41 ( or both ) is diffeomorphic to a trivial block @xmath117 . [ irred_def_2 ] let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath115}$ ] and transverse to @xmath31 . assume @xmath50 meets each component of @xmath31 at exactly one point . the block @xmath39 is * irreducible * provided : there is a neighborhood @xmath235 of @xmath236}$ ] in @xmath115}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath115}$ ] such that :    1 . [ h0id ] @xmath177 . [ hrelu ] @xmath238 for all @xmath176 . [ r_setwise ] @xmath175 fixes @xmath31 setwise for all @xmath237 . 4 .   @xmath239 is a level @xmath17-sphere in @xmath115}$ ] . @xmath263 has no critical points between @xmath239 and @xmath113}$ ] , or between @xmath239 and @xmath233}$ ] . [ defs_equiv ] the two definitions of irreducible block are equivalent . assume @xmath39 is irreducible according to the second definition . suppose @xmath264 is a diffeomorphism . assume @xmath220 sends the inner boundary @xmath17-sphere to the inner boundary @xmath17-sphere , the other case being similar . let @xmath252 be the level @xmath17-sphere in @xmath40 along which the concatenation takes place . let @xmath265 . then , @xmath50 satisfies the hypotheses of the second definition . let @xmath175 be the isotopy provided by the second definition . so , @xmath239 is a level @xmath17-sphere in @xmath115}$ ] and , say , @xmath263 has no critical points in the compact region @xmath259 with boundary @xmath266}$ ] . by corollary  [ trivial_block_cor ] , the block @xmath267 is diffeomorphic to a trivial block @xmath117 . hence , @xmath268 , as desired . + next , assume @xmath39 is irreducible according to the first definition . let @xmath50 satisfy the hypotheses of the second definition . by lemma  [ ess_sphere ] , @xmath50 is essential in @xmath115}$ ] , and there is an isotopy of @xmath115}$ ] ( probably disturbing @xmath31 ) which carries @xmath50 to a level @xmath17-sphere , @xmath252 . let @xmath269 be the image of @xmath31 under this isotopy . this @xmath252 divides @xmath116},\tau')$ ] into two obvious blocks @xmath26 and @xmath41 . evidently , @xmath39 is diffeomorphic to the concatenation @xmath40 . the first definition of irreducible block implies that , say , @xmath26 is diffeomorphic to @xmath117 . by lemma  [ en_unique ] ( trivial block uniqueness ) , we can and do assume @xmath117 has underlying thickened sphere @xmath247 $ ] . let @xmath259 denote the compact region in @xmath115}$ ] between @xmath50 and @xmath113}$ ] . hence , there is a diffeomorphism of pairs @xmath270 , and @xmath197 sends @xmath113}$ ] to @xmath146 . let @xmath271}$ ] for some small @xmath272 to be specified . first , choose @xmath135 small enough so @xmath273 lies between @xmath50 and @xmath113}$ ] . next , reduce @xmath135 if necessary so that :    1 . [ nocp ] @xmath263 has no critical points on or between @xmath273 and @xmath113}$ ] . this reduction is possible since @xmath31 is neatly embedded . note that futher reducing @xmath135 maintains condition  [ nocp ] . finally , reduce @xmath135 if necessary so that :    1 . [ nottangent ] @xmath274 is nowhere tangent to any radial arc in @xmath21}$ ] . this last reduction is possible since : ( i ) @xmath197 is a diffeomorphism , ( ii ) @xmath275})=s^2 $ ] , and ( iii ) @xmath113}$ ] is compact . as @xmath274 is essential in @xmath21}$ ] , condition  [ nottangent ] implies that :    1 . [ onepointtransverse ] each radial arc in @xmath21}$ ] of length @xmath276 intersects @xmath274 in exactly one point and transversely . condition  [ onepointtransverse ] permits construction of an ambient isotopy of @xmath21}$ ] that carries @xmath274 to @xmath277}$ ] and merely slides points along radial arcs . so , by an abuse of notation , we further assume the diffeomorphism of pairs @xmath197 itself sends @xmath273 to @xmath277}$ ] . let @xmath278 be the compact region in @xmath115}$ ] between @xmath50 and @xmath273 . construct a vector field @xmath190 on @xmath115}$ ] as follows . on @xmath278 , @xmath190 is the pushforward by @xmath260 of the vector field @xmath279 on @xmath280}$ ] . note that @xmath190 is tangent to @xmath31 on @xmath278 . extend @xmath190 to the rest of @xmath115}$ ] , making it @xmath98 outside a small neighborhood of @xmath278 and ensuring tangency to @xmath31 . the isotopy generated by @xmath190 is the desired @xmath175 .    in general , it appears to be a difficult problem to decide whether a given block is irreducible . we prove next that trivial blocks are irreducible and then observe some corollaries . [ sigma_en ] each trivial block @xmath168},\tau)$ ] , @xmath110 , is irreducible . let @xmath281 denote the annulus where @xmath21}$ ] meets the @xmath123-plane . lemma  [ en_unique ] reduces us to the case where @xmath117 has underlying thickened sphere @xmath21}$ ] and @xmath31 consists of @xmath37 equally spaced radial arcs in @xmath281 . we prove @xmath117 is irreducible according to the first definition . by the first paragraph of the proof of proposition  [ defs_equiv ] , it suffices to consider a @xmath17-sphere , @xmath50 , embedded in the interior of @xmath21}$ ] , transverse to @xmath31 , and intersecting each component @xmath61 of @xmath31 in one point @xmath70 . let @xmath282}$ ] be the compact set between @xmath50 and @xmath146 . it suffices to produce a diffeomorphism @xmath283 . + we will improve @xmath50 ( and , hence , @xmath284 ) by ambient isotopies of @xmath21}$ ] . improved spaces will be denoted by their original names , except @xmath281 always denotes @xmath21 } \cap { \textnormal{($xy$-plane)}}$ ] . so , assume @xmath50 intersects @xmath281 transversely . thus , @xmath285 is a closed @xmath276-manifold and one component , @xmath133 , of @xmath285 must contain all of the points @xmath70 . + if @xmath286 , then consider a component , @xmath134 , of @xmath287 that is innermost in its component of @xmath288 . let @xmath289 be the @xmath17-disk in @xmath288 with boundary @xmath134 . let @xmath290 be the @xmath17-disk in @xmath281 with boundary @xmath134 . then , @xmath291 is an embedded @xmath17-sphere in the interior of @xmath21}$ ] disjoint from @xmath31 . by lemma  [ ess_sphere ] , @xmath291 is inessential in @xmath21}$ ] . let @xmath77 be the @xmath27-disk in @xmath21}$ ] with boundary @xmath291 . this @xmath77 permits construction of an isotopy of @xmath21}$ ] , with support near @xmath77 , that carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath292 . repeating this operation finitely many times , we get @xmath293 . + now , we give a bootstrapping definition of the required diffeomorphism @xmath197 . first , @xmath197 sends @xmath294 by the identity . second , @xmath197 sends @xmath295 to @xmath61 by an affine diffeomorphism for each @xmath296 . third , @xmath197 sends @xmath133 to @xmath253}\cap\pi$ ] . fourth , @xmath197 sends a smooth , regular neighborhood of @xmath297 in @xmath298 to a smooth , regular neighborhood of @xmath299}\cap\pi\right)}$ ] in @xmath281 . this step may be accomplished , quite concretely , by judiciously choosing ( closed ) collars  @xcite and ambiently rounding corners . fifth , @xmath197 sends @xmath298 to @xmath281 . by the smooth @xmath17-dimensional schoenflies theorem ( * remark  9.19 ) , this step evidently requires extension of @xmath197 over @xmath37 smooth @xmath17-disks @xmath300 , @xmath301 . let @xmath302 denote the @xmath17-disk with boundary @xmath303 . as every diffeomorphism of @xmath71 extends to one of @xmath304 , each diffeomorphism @xmath305 extends to a diffeomorphism @xmath306 . extending @xmath197 over @xmath307 by @xmath308 yields a well - defined homeomorphism ( smooth except possibly at @xmath309 ) . by an isotopy of @xmath308 , relative to @xmath309 and with support in a collar of @xmath309  @xcite , this extension is a diffeomorphism . sixth , @xmath197 sends @xmath310}$ ] . seventh , @xmath197 sends a smooth , regular neighborhood of @xmath311 in @xmath284 to a smooth , regular neighborhood of @xmath312}$ ] in @xmath21}$ ] . this is done as in step four ( product a nice corner rounding with @xmath71 ) . finally , @xmath197 sends @xmath284 to @xmath21}$ ] . by the smooth @xmath27-dimensional schoenflies theorem , this step requires extension of @xmath197 over two smooth @xmath27-disks . this is done as in step five , except using the fact that every diffeomorphism of @xmath146 extends to one of @xmath313 @xcite , @xcite , @xcite . by construction , the diffeomorphism @xmath314}$ ] sends @xmath315 to @xmath31 . [ cor1 ] let @xmath23 , @xmath316 , be any blocks . if @xmath317 , then @xmath318 for each @xmath316 .    by induction , it suffices to consider the case @xmath319 . the case @xmath319 follows from the proof of proposition  [ sigma_en ] , since @xmath284 can be the compact region between @xmath50 and @xmath146 , or between @xmath50 and @xmath253}$ ] . alternatively , it is instructive to see that the case @xmath319 follows from the statement of proposition  [ sigma_en ] as follows . proposition  [ sigma_en ] implies that @xmath268 or @xmath320 . assume @xmath268 ( the other case is similar ) . we have diffeomorphisms : @xmath321 where the first exists by hypothesis , the second follows from lemma  [ diff_impl_diff ] since @xmath268 and @xmath322 , and the last is given by lemma  [ e_nidentity ] . [ cor2 ] let @xmath23 , @xmath148 , be any blocks . if @xmath323 , then @xmath324 for each @xmath325 . let @xmath326 be a diffeomorphism ( of pairs ) . let @xmath31 and @xmath8 be the @xmath37-component multirays determined by these concatenations respectively . clearly , @xmath8 is radial . isotopies of @xmath35 will send @xmath31 to @xmath8 at all times . note that @xmath327 since @xmath35 must restrict to a diffeomorphism on the boundaries of the total spaces . let @xmath328}\right)}$ ] . by a radial isotopy , relative to @xmath146 , we may assume @xmath329}}$ ] . this @xmath50 satisfies the hypotheses in the second definition of irreducible block . propositions  [ defs_equiv ] and  [ sigma_en ] permit us to isotop @xmath35 , relative to @xmath146 , so that @xmath330}$ ] . having isotoped @xmath35 so that @xmath331}\right)}=s^2{\left[i\right]}$ ] for @xmath316 and some @xmath332 , the same argument permits us to further isotop @xmath35 , relative to @xmath333}$ ] , so that @xmath334}\right)}=s^2{\left[k+1\right]}$ ] . evidently , the composition of all of these ( infinitely many ) isotopies is a well - defined , smooth , proper isotopy . so , we can and do assume @xmath335}\right)}=s^2{\left[k\right]}$ ] for all @xmath336 . the result is now immediate . [ cor3 ] let @xmath23 , @xmath148 , be any blocks . if @xmath337 , then @xmath324 for all sufficiently large @xmath24 . let @xmath338 be a diffeomorphism ( of pairs ) . let @xmath31 and @xmath8 be the @xmath37-component multirays determined by these concatenations respectively . isotopies of @xmath35 will send @xmath31 to @xmath8 at all times . by compactness , there exists @xmath332 such that @xmath335}\right)}$ ] is disjoint from @xmath313 . as in the previous proof , we may isotop @xmath35 , relative to @xmath313 , so that @xmath335}\right)}=s^2{\left[k\right]}$ ] . restricting @xmath35 to @xmath339 implies @xmath340 . now , apply the previous corollary . borromean blocks were defined in section  [ borr_blocks ] . a multiray @xmath30 forms * borromean rays * provided : ( i ) no diffeomorphism of @xmath0 carries @xmath31 to a radial multiray , and ( ii ) each multiray obtained from @xmath31 by forgetting one component is ambiently isotopic to a two component , radial multiray . [ borr_blocks_yield_borr_rays ] let @xmath23 , @xmath148 , be borromean blocks and let : @xmath341 then , @xmath31 forms borromean rays . in particular , the conclusion holds if each @xmath342 .    by definition of borromean block , corollary  [ cor3 ] implies that @xmath344 . hence , no diffeomorphism of @xmath0 carries @xmath31 to a radial multiray . next , let @xmath32 be obtained from @xmath31 by forgetting any one component . let @xmath33 be obtained from @xmath23 by forgetting the corresponding tangle component . by definition of borromean block , each @xmath33 is ambient isotopic ( relative to boundary ) to @xmath157 . performing these isotopies , for @xmath24 , simultaneously yields an ambient isotopy of @xmath0 carrying @xmath32 to a radial multiray . thus , @xmath31 forms borromean rays . lastly , blocks in @xmath18 are borromean by corollary  [ four_blocks_borromean ] .    1 . corollary  [ cor3 ] reduces the _ infinite _ problem of constructing knotted multirays to the _ finite _ problem of constructing nontrivial blocks . for instance , the infinitely generated group theory in  @xcite may be replaced by finitely generated group theory ( as used in section  [ equiv_rel_blocks ] above ) . 2 .   the converse of corollary  [ cor3 ] holds by remarks  [ concat_remarks ] item  [ tail_det_type ] . we close this section by constructing infinitely many irreducible blocks containing two component tangles . recall the notion of a knot block from example  [ knot_blocks ] . [ pk_ib ] if @xmath151 is a prime knot , then @xmath156 is an irreducible block . let @xmath345},\tau\right)}$ ] where @xmath43 is radial and @xmath42 contains the diagram @xmath133 for @xmath160 as in figure  [ knot_block ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath21}$ ] , transverse to @xmath31 , and intersecting each component @xmath61 of @xmath31 in one point @xmath70 . perturb @xmath50 so it coincides with the level sphere through @xmath217 near @xmath43 . all isotopies will be ambient and relative to a neighborhood of both @xmath346 $ ] and @xmath43 . subsets that move will be called by their original names . as in the proof of proposition  [ sigma_en ] , we isotop @xmath50 to the level sphere containing @xmath217 . push @xmath42 away from @xmath43 by integrating a vector field tangent to level spheres . the result is shown in figure  [ knot_block_comp ] . after ambient isotopy carrying @xmath50 to a level sphere . ]    as @xmath160 is prime , one of the diagrams @xmath347 or @xmath348 must be trivial . [ inf_ib ] there exists a countably infinite collection of irreducible knot blocks , pairwise distinct up to diffeomorphism . let @xmath349 denote the set of torus knots @xmath350 where @xmath351 and @xmath352 . as torus knots are prime  @xcite , lemma  [ pk_ib ] implies that each @xmath156 , @xmath353 , is irreducible . the fundamental groups of these torus knots are pairwise nonisomorphic  @xcite . by example [ knot_blocks ] , these knot blocks are pairwise distinct up to diffeomorphism .    under concatenation , knot blocks commute , unlike distinct blocks in @xmath18 ( recall table  [ magma_data_2 ] ) . concatenating infinitely many knot blocks yields a multiray in @xmath0 known as _ wilder rays_. they were classified by fox and harrold  @xcite . this section identifies some unknotted ball - arc pairs in blocks . these tools will be used in the next two sections . recall that a * ball - arc pair * is a pair @xmath354 such that @xmath355 is neatly embedded in @xmath356 . such a pair is * unknotted * provided it is diffeomorphic to the standard pair @xmath357 , and otherwise it is * knotted*. if @xmath151 is a smooth knot ( not the unknot ) and @xmath358 is an unknotted ball - arc pair such that @xmath359 , then @xmath360 is a knotted ball - arc pair . every knotted ball - arc pair arises this way up to diffeomorphism . +    [ ball - arc ] let @xmath361},\tau\right)}$ ] . let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath21}$ ] and transverse to @xmath31 . assume @xmath50 meets @xmath31 at exactly two points @xmath10 and @xmath137 , both of which lie on one component of @xmath31 , say @xmath42 . then , @xmath50 bounds a @xmath27-disk , @xmath246 , in @xmath21}$ ] and @xmath362 is an unknotted ball - arc pair .    by lemma  [ ess_sphere ] , @xmath50 is inessential in @xmath21}$ ] . let @xmath281 denote the annulus where @xmath21}$ ] meets the @xmath123-plane . without loss of generality , @xmath363 and @xmath50 is transverse to @xmath281 . thus , @xmath285 is a closed @xmath276-manifold and one component , @xmath133 , of @xmath285 must contain @xmath10 and @xmath137 ; this is where the hypothesis @xmath364 is used . as in the proof of proposition  [ sigma_en ] ( paragraph three ) , we may arrange that @xmath365 . now , it is straightforward to construct the required diffeomorphism ( cf . paragraph four of the proof of proposition  [ sigma_en ] ) .    1 . lemma  [ ball - arc ] becomes false without the hypothesis @xmath364 ( i.e. , with @xmath157 replaced by @xmath132 ) . to see this , consider the block @xmath131},\tau'\right)}$ ] in figure  [ sphere_two_points ] ( left ) . +   meeting the tangle @xmath269 at two points in the block @xmath366 . at right is the result of an ambient isotopy that fixes @xmath269 setwise . ] + the indicated sphere @xmath252 meets @xmath269 in two points and bounds the @xmath27-ball @xmath367 . let @xmath133 be any crossing diagram such that @xmath368 is a knotted ball - arc pair . straighten @xmath269 using the argument in example  [ one_comp ] . let @xmath31 , @xmath50 , and @xmath246 denote the respective images of @xmath269 , @xmath252 , and @xmath367 under this ambient isotopy . then , @xmath50 is a @xmath17-sphere in @xmath132 meeting @xmath31 in exactly two points and transversely . however , @xmath369 is a knotted ball - arc pair . 2 .   lemma  [ ball - arc ] and the previous remark may be recast in @xmath0 as follows . consider a @xmath17-sphere @xmath370 . let @xmath371 be the @xmath27-disk with @xmath372 . suppose @xmath13 is a straight arc in @xmath0 that is neatly embedded in @xmath246 . let @xmath373 be the straight line containing @xmath13 . if @xmath50 is disjoint from @xmath374 , then @xmath354 is an unknotted ball - arc pair . if @xmath50 meets @xmath374 , then @xmath354 may be a knotted ball - arc pair . in fact , every knotted ball - arc pair @xmath375 in @xmath0 is ambient isotopic to some such @xmath354 . proof : ( i ) straighten @xmath376 near an endpoint @xmath377 , ( ii ) let @xmath378 be a point in the straightened end of @xmath376 , and ( iii ) ambiently isotop the other endpoint of @xmath376 along @xmath376 until it concides with @xmath379 ( cf . figure  [ sphere_two_points ] ( right ) ) . @xmath150    [ disk_ess_sphere ] let @xmath13 be a neatly embedded arc in @xmath115}$ ] . assume @xmath13 has one boundary point in @xmath113}$ ] and the other in @xmath233}$ ] . let @xmath289 be a @xmath17-disk embedded in the interior of @xmath115}$ ] . assume that @xmath380 lies in some @xmath381}$ ] , @xmath134 is disjoint from @xmath13 , @xmath382 is disjoint from @xmath381}$ ] , and @xmath289 is transverse to @xmath381}$ ] . let @xmath290 and @xmath383 be the two @xmath17-disks in @xmath381}$ ] bounded by @xmath134 . then :    1 . [ indneq ] the intersection numbers @xmath384 and @xmath385 are unequal . 2 .   the intersection numbers @xmath386 and @xmath384 are equal ( after possibly interchanging the names of @xmath290 and @xmath383 ) . the sphere @xmath291 is inessential in @xmath115}$ ] . the sphere @xmath387 is essential in @xmath115}$ ] . immediate by lemma  [ ess_sphere ] . let @xmath388 denote closed upper half space . the closed upper half disk is @xmath389 . [ ball - arc_diffeo ] let @xmath361},\tau\right)}$ ] . let @xmath42 be a component of @xmath31 and let @xmath390}$ ] . suppose @xmath289 is a @xmath17-disk neatly embedded in @xmath21}$ ] such that : ( i ) @xmath289 is transverse to @xmath31 , ( ii ) @xmath289 meets @xmath31 at one point @xmath391 , and ( iii ) @xmath392 lies in @xmath253}$ ] . let @xmath290 be the @xmath17-disk in @xmath253}$ ] with boundary @xmath134 and containing @xmath137 . then , @xmath393 bounds a piecewise smooth @xmath27-disk @xmath394}$ ] and @xmath395 . further , there is a diffeomorphism of pairs @xmath396 that sends @xmath289 to the upper hemisphere and @xmath290 to @xmath397 .    by lemma  [ disk_ess_sphere ] with @xmath398 , @xmath393 is inessential in @xmath21}$ ] . by hypothesis , so , @xmath400 , and @xmath395 as well . the required diffeomorphism @xmath197 is constructed in bootstrapping fashion ( cf . paragraph four of the proof of proposition  [ sigma_en ] ) : define @xmath197 on @xmath401 , extend to a smooth , regular neigborhood of @xmath401 in @xmath77 , and extend to the rest of @xmath77 utilizing lemma  [ ball - arc ] . [ a_irred ] each block in @xmath18 is irreducible . the remainder of this section is devoted to proving @xmath19 is irreducible , which suffices to prove theorem  [ a_irred ] . recall the block @xmath402},t\right)}$ ] from figure  [ four_blocks ] . the ( general position ) immersion @xmath35 yielding @xmath19 ( see figure  [ immersion ] ) plays a central role in our proof . we pause to explain @xmath35 and fix some notation . we assume the reader has figure  [ immersion ] at hand . + the domain of @xmath35 is @xmath403}\subset\r^2 $ ] . the compact annulus @xmath73 contains three equally spaced radial arcs , @xmath42 , @xmath43 , and @xmath404 , as in figure  [ immersion_domain ] .     of the immersion @xmath35 . ]    let @xmath405 . note that @xmath406 , the tangle in @xmath19 . we let @xmath407}$ ] , the subannulus of @xmath73 that is unshaded in figure  [ immersion_domain ] . the boundary of @xmath408 is @xmath409 , the disjoint union of @xmath410}$ ] and @xmath411}$ ] . + given a subset @xmath412 , it will be convenient to let @xmath413 denote @xmath414 . ( a notable exception is @xmath415 . ) in particular , @xmath416 , @xmath417 , and @xmath418 . + for each @xmath419}$ ] , @xmath420}$ ] is an embedding , namely the composition of : a rigid rotation , a homothety , and a translation in the @xmath120-direction . we will see that :    1 . the multiple points of @xmath35 are double points where @xmath421}\right)}=f{\left(s^1{\left[\sfrac{5}{3}\right]}\right)}=\omega'$ ] . 2 .   @xmath422}\times{\left\{\varepsilon\right\}}$ ] where @xmath423 . 3 .   @xmath424 is a torus in @xmath21}$ ] , smooth except for corners along @xmath425 . on @xmath426 , @xmath35 is inclusion @xmath427 .     with plane containing @xmath120-axis . ]    figure  [ slice ] shows the intersection of @xmath428 with any plane in @xmath0 containing the @xmath120-axis . on each of the three subannuli of @xmath73 in figure  [ immersion_domain ] , @xmath35 is defined as follows . the annulus @xmath429}$ ] is stretched radially to @xmath430}$ ] , then the outer boundary component is twisted by @xmath431 radians ccw while fixing the inner boundary component , then level circles near the outer boundary component are lifted up a bit in the @xmath120-direction ( to yield general position ) . the annulus @xmath432}$ ] maps into @xmath21}$ ] by @xmath433 , then level circles near the inner boundary component are lifted up a bit in the @xmath120-direction ( again , to yield general position ) . finally , @xmath35 is defined on @xmath408 , interpolating @xmath434}$ ] and @xmath435}$ ] , so as to yield a torus @xmath436 as in figures  [ immersion ] and  [ slice ] . the two components of @xmath437 are identified under @xmath35 after half a rotation of @xmath410}$ ] . this completes our description of @xmath35 . + for distinct @xmath438 , let @xmath68 denote the closed sector in @xmath73 between @xmath61 and @xmath63 of angular measure @xmath439 . note that @xmath440 is an embedding . fix distinct @xmath441 . observe that @xmath442 meets @xmath443 ( transversely ) at exactly two points . for example , using the labelings in figure  [ immersion_domain ] , @xmath444 meets @xmath445 at the two points : @xmath446 similarly , @xmath447 and @xmath448 for @xmath449 and @xmath17 . the points @xmath450 , @xmath451 , and @xmath452 , where @xmath449 and @xmath17 , will be referred to as * special points*. + we prove @xmath19 is irreducible according to definition  [ irred_def_1 ] . it suffices to consider a @xmath17-sphere , @xmath50 , embedded in @xmath453}$ ] , transverse to @xmath20 , and meeting each component of @xmath20 at exactly one point . we improve @xmath50 by ambient isotopies of @xmath21}$ ] that fix @xmath20 setwise at all times . by an abuse , we refer to each improved @xmath50 as @xmath50 . we view @xmath416 as an auxiliary object , unaffected by these isotopies . perturb @xmath50 so that @xmath454 is disjoint from @xmath425 . perturb @xmath50 again so that further @xmath50 meets @xmath428 in general position . in particular , @xmath455 is an immersed , closed @xmath276-manifold in @xmath50 in general position . define : @xmath456 which is an embedded , closed @xmath276-manifold in @xmath457 , transverse to @xmath437 and @xmath31 . each component , @xmath61 , of @xmath31 meets @xmath32 at exactly one point ( not in @xmath437 ) . so , there exists one component , @xmath133 , of @xmath32 that meets each @xmath61 at one point ( transversely ) and @xmath133 is essential in @xmath73 . [ suff_claim ] it suffices to arrange that @xmath458 and @xmath459 . similar to the argument in paragraph four of the proof of proposition  [ sigma_en ] . we give three operations for improving @xmath50 . define the * complexity * of @xmath50 to be : @xmath460    . ] [ taco0 ] suppose that @xmath461 and @xmath462 are arcs , and @xmath463 is a simple closed curve bounding a disk @xmath464 . assume that @xmath77 contains no special points , @xmath465 , and @xmath466 ( see figure  [ taco_hypo ] ) . then , the points of @xmath467 can be eliminated from @xmath468 , and @xmath469 decreases by at least @xmath27 .    as @xmath77 intersects only one component of @xmath437 , @xmath470 is an embedding . the disk @xmath471 permits construction of an isotopy of @xmath50 , with support near @xmath158 , that carries @xmath472 past @xmath473 to a parallel copy of @xmath473 . if @xmath158 intersects @xmath20 , then @xmath474 is a disk - arc pair ( all of which are unknotted ) . so , @xmath475 is an unknotted disk - arc pair , and the isotopy fixes @xmath20 setwise . the reduction in @xmath469 follows from figure  [ taco_complexity ] .     of @xmath476}$ ] correspond to two points of @xmath477}$ ] . ] four points of @xmath468 are eliminated , and , at worst , the number of components of @xmath32 increases by one . [ inessential ] among the components of @xmath32 that are inessential in @xmath73 and disjoint from @xmath437 , let @xmath134 be one that is innermost in @xmath73 . then , @xmath134 can be eliminated from @xmath32 , and @xmath469 decreases by at least @xmath276 . let @xmath464 be the @xmath17-disk with @xmath478 . note that @xmath77 is disjoint from @xmath437 and @xmath31 . so , @xmath470 is an embedding and @xmath471 is a @xmath17-disk disjoint from @xmath20 and bounding @xmath479 . the circle @xmath480 bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . the arc @xmath444 meets @xmath50 at one point . so , without loss of generality , @xmath444 meets @xmath289 at one point ( transversely , and in @xmath481 ) , and @xmath482 . by hypothesis , so , @xmath484 and @xmath485 are embedded @xmath17-spheres . by lemma  [ ess_sphere ] , @xmath486 is essential in @xmath21}$ ] and @xmath485 is inessential . so , @xmath485 bounds an embedded @xmath27-disk @xmath487}$ ] , and @xmath488 . the @xmath27-disk @xmath246 permits construction of an isotopy of @xmath50 , with support near @xmath246 , that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . [ one_special_point ] let @xmath134 be a component of @xmath32 that bounds a @xmath17-disk @xmath464 . assume that @xmath489 is a neatly embedded arc in @xmath77 , @xmath77 contains exactly one special point @xmath118 , @xmath490 , and @xmath491 . then , @xmath134 can be eliminated from @xmath32 , and @xmath469 decreases by at least @xmath492 .    without loss of generality , @xmath493 as in figure  [ ball - arc - push ] . correspond to two points of @xmath477}$ ] . ] the embedded @xmath17-disk @xmath471 meets @xmath20 at exactly @xmath494 . the circle @xmath479 bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . without loss of generality , @xmath495 and @xmath444 meets @xmath290 at one point ( transversely ) . by lemma  [ ess_sphere ] , @xmath485 bounds an embedded @xmath27-disk @xmath496}$ ] and @xmath497 is a neatly embedded arc in @xmath246 . by lemma  [ ball - arc ] , @xmath498 is an unknotted ball - arc pair . the pair @xmath498 permits construction of an isotopy of @xmath50 , with support near @xmath246 , that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . this isotopy fixes @xmath444 setwise and fixes @xmath499 and @xmath500 pointwise . improve @xmath50 by applying lemmas  [ taco0 ] , [ inessential ] , and [ one_special_point ] _ in any order and as long as possible_. this is a finite process since the initial complexity of @xmath50 is a positive integer and each operation strictly reduces the complexity . the complexity of the resulting improved @xmath50 is @xmath501 . the rest of this section shows that @xmath502 and @xmath459 , which suffices to prove theorem  [ a_irred ] by claim  [ suff_claim ] . [ taco ] there do not exist arcs @xmath461 and @xmath462 such that @xmath463 is a simple closed curve bounding a disk @xmath464 where @xmath77 is disjoint from the special points . we have @xmath503 where @xmath39 is a finite disjoint union of neatly embedded arcs in @xmath77 . we claim that @xmath254 contains no closed component of @xmath32 . otherwise , let @xmath504 be the @xmath17-disk bounded by an innermost such component @xmath134 . if @xmath505 , then lemma  [ inessential ] applies to @xmath134 , a contradiction . thus , @xmath506 is a nonempty finite disjoint union of neatly embedded arcs in @xmath246 . let @xmath507 be an arc of @xmath506 that is outermost in @xmath246 in the sense that @xmath508 subtends an arc @xmath509 such that : @xmath510 lemma  [ taco0 ] applies to @xmath511 and @xmath507 , a contradiction . the proof of the claim is complete . + if @xmath512 , then lemma  [ taco0 ] applies to @xmath13 and @xmath513 , a contradiction . otherwise , there exists a component @xmath507 of @xmath39 that is outermost in @xmath77 in the sense that @xmath508 subtends an arc @xmath514 such that : @xmath515 lemma  [ taco0 ] applies to @xmath511 and @xmath507 , a contradiction . next , we show that @xmath32 contains no component inessential in @xmath73 . suppose , by way of contradiction , that @xmath32 contains component(s ) inessential in @xmath73 . among these components , there must be one , call it @xmath134 , that is innermost in @xmath73 . by lemma  [ inessential ] , @xmath134 meets @xmath437 . so , @xmath516 is positive and even . let @xmath464 be the @xmath17-disk with @xmath478 . note that @xmath517 . also , @xmath518 is a nonempty , finite disjoint union of neatly embedded arcs in @xmath77 . recall that the only component of @xmath32 that meets @xmath31 is @xmath133 , and @xmath133 is essential in @xmath73 . so , @xmath77 is contained in the interior of a sector @xmath68 and @xmath470 is an embedding . without loss of generality , assume @xmath519 . let @xmath471 , an embedded @xmath17-disk with @xmath520 . the circle @xmath480 also bounds two @xmath17-disks , @xmath289 and @xmath290 , in @xmath50 . as @xmath521 , @xmath486 and @xmath485 are embedded @xmath17-spheres in @xmath21}$ ] . by lemma  [ ess_sphere ] ( using the arc @xmath444 , say ) , one of these spheres is essential in @xmath21}$ ] and the other is inessential . without loss of generality , assume @xmath486 is essential in @xmath21}$ ] and @xmath485 is inessential . let @xmath246 be the @xmath27-disk in @xmath21}$ ] with @xmath522 . [ no_inessential ] the disk @xmath77 can not be disjoint from the special points @xmath523 and @xmath524 . otherwise , let @xmath513 be a component of @xmath39 . let @xmath525 be an arc with @xmath526 . the arcs @xmath13 and @xmath513 contradict lemma  [ taco ] . [ 2special ] the disk @xmath77 can not contain both special points @xmath523 and @xmath524 . suppose otherwise . note that @xmath444 meets @xmath158 twice ( transversely in @xmath527 ) , and @xmath499 and @xmath500 are disjoint from @xmath158 . by lemma  [ ess_sphere ] , @xmath20 is disjoint from @xmath290 . the disk @xmath246 permits construction of an an ambient isotopy @xmath528 , @xmath176 , of @xmath21}$ ] that carries @xmath290 past @xmath158 to a parallel copy of @xmath158 . this isotopy has support near @xmath246 , is relative to a neighborhood of @xmath529}$ ] , but does _ not _ fix @xmath444 setwise . note that @xmath530 where @xmath531 still denotes @xmath532 . as @xmath533 is a borromean block , there is an ambient isotopy @xmath534 , @xmath535 , of @xmath21}$ ] , relative to a neighborhood of @xmath536}$ ] , that straightens both @xmath537 and @xmath538 . the strip @xmath539 permits construction of an ambient isotopy @xmath175 , @xmath176 , of @xmath21}$ ] that carries @xmath540 to an arc close to and winding around @xmath541 . this isotopy has support near @xmath539 , is relative to both @xmath542 and @xmath541 , but otherwise is not relative to @xmath536}$ ] . the resulting block @xmath543 is diffeomorphic to @xmath165 by untwisting tangle components @xmath17 and @xmath27 . so , @xmath544 , which contradicts corollary  [ anottrivial ] . [ one ] the disk @xmath77 can not contain exactly one of the special points @xmath523 or @xmath524 . assume , without loss of generality , that @xmath254 contains @xmath523 but not @xmath524 . recall that @xmath517 . if @xmath39 is connected , then lemma  [ one_special_point ] applies to @xmath77 , a contradiction . if @xmath39 is disconnected , then @xmath39 contains a component , @xmath513 , that does not contain @xmath523 . let @xmath525 be the arc with @xmath526 and such that @xmath523 does not lie inside the simple closed curve @xmath463 . the arcs @xmath13 and @xmath513 contradict lemma  [ taco ] . taking stock , @xmath32 contains no component inessential in @xmath73 . so , @xmath458 is a single , essential circle in @xmath73 , and @xmath133 meets each @xmath61 at one point ( transversely ) . it remains to prove that @xmath459 . a * segment * will mean an arc @xmath545 for which @xmath546 . each inline figure , such as , represents a segment of @xmath133 in a sector ; vertical lines represent adjacent components of @xmath31 , upper and lower horizontal lines represent arcs of @xmath437 , dots represent special points , and arrows indicate reflected cases of the entire figure . for example ,   represents six cases ( three choices of sector and a possible vertical reflection ) , and   represents twelve cases . [ scin ] none of the following appear : , , or . observe that points of @xmath73 inside ( outside ) @xmath133 map under @xmath35 to lie inside ( outside ) @xmath50 respectively . suppose there is . without loss of generality , assume the indicated special points are @xmath523 and @xmath524 . then , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 . there are three possibilities for the location of @xmath549 ( see figure  [ annulus_omo ] at right ) . the inner option implies @xmath550 and @xmath551 both lie outside @xmath50 . the middle option implies @xmath550 lies inside @xmath50 and @xmath551 lies outside @xmath50 . the outer option implies @xmath550 and @xmath551 both lie inside @xmath50 . all three are contradictions since @xmath552 and @xmath553 . so , no   appears . we claim that :    1 . [ not_both ] there do not exist   and   in the same sector and with the same reflection .    to see this , suppose , by way of contradiction , that   and   both appear in the same sector , say @xmath554 . let @xmath555 and @xmath556 denote these segments respectively . assume , without loss of generality , that the special point just below @xmath555 is @xmath523 . let @xmath557 be the unique arc such that @xmath558 and @xmath559 . lemma  [ taco ] implies that @xmath560 bounds segments @xmath561 parallel to @xmath555 and @xmath562 parallel to @xmath556 ( @xmath563 and @xmath564 are possible ) . then , @xmath565 is an inessential component of @xmath32 , a contradiction . this completes our proof of  [ not_both ] . + suppose there is . assume , without loss of generality , that this segment @xmath566 of @xmath133 lies in @xmath554 . by lemma  [ taco ] , the only possible segment of @xmath133 that meets both @xmath408 and @xmath567 , and is disjoint from @xmath31 , is . by  [ not_both ] , the boundary points of @xmath566 lie in distinct segments , @xmath568 and @xmath569 , of @xmath133 , where @xmath568 meets @xmath43 at one endpoint and @xmath569 meets @xmath404 at one endpoint ( see figure  [ annulus_omo ] ) .    . ]    so , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 . this yields the same contradiction as for   above . hence , no   appears . + suppose there is . call this segment @xmath566 and assume , without loss of generality , that @xmath523 is the special point pictured above @xmath566 . let @xmath557 denote the short arc with @xmath570 ( see figure  [ annulus_omm ] ) . we claim that no segment of @xmath133 contained in @xmath408 may meet @xmath120 . this follows from : ( i ) lemma  [ taco ] , ( ii ) the nonexistence of , ( iii ) since @xmath566 meets @xmath43 , and ( iv ) since @xmath133 meets @xmath43 exactly once . therefore , @xmath571 appears as in figure  [ annulus_omm ] .     can not exit the annulus @xmath408 between @xmath572 and @xmath523 . ] so , @xmath547 is outside @xmath50 and @xmath548 is inside @xmath50 , a familiar contradiction . hence , no   appears . recall that @xmath573}$ ] is an embedded torus , smooth except for corners along @xmath425 . so , @xmath574 is a closed @xmath276-manifold , smooth except for corners , and embedded in @xmath50 and in @xmath424 . we introduce a based longitude @xmath575 and a based meridian @xmath576 on @xmath424 . both originate and terminate at @xmath577 ( recall figures  [ immersion ] , [ immersion_domain ] , and [ slice ] ) . the longitude @xmath575 runs once along @xmath425 in the cw direction about the @xmath120-axis when viewed from the point @xmath578 . the meridian @xmath576 is the right loop in figure  [ slice ] , oriented ccw , where the plane of intersection is the @xmath579-plane . in particular , a parallel pushoff @xmath580 of @xmath575 into the inside of @xmath424 has linking number @xmath198 with @xmath576 in @xmath0 . an oriented loop in @xmath424 has * type * @xmath581 provided it is freely homotopic in @xmath424 to @xmath582 . [ essential ] let @xmath134 be a component of @xmath574 , equipped with an orientation and of type @xmath581 . then , @xmath583 . assume @xmath134 is essential in @xmath424 ( otherwise , the result is clear ) . note that @xmath584 . focus attention on the submanifolds @xmath50 and @xmath424 of @xmath21}\subset\r^3 $ ] . we view @xmath50 and @xmath424 as submanifolds of @xmath585 , where @xmath586 and @xmath587 . as @xmath134 is essential in @xmath424 , @xmath134 is not null - homologous ( denoted @xmath588 ) in both @xmath589 and @xmath590 . let @xmath284 denote @xmath589 or @xmath590 where @xmath588 in @xmath284 . exactly one component of @xmath591 contains @xmath134 in its frontier and lies in @xmath284 ; let @xmath592 denote the closure of this component in @xmath50 . note that @xmath592 is a compact @xmath17-disk with holes , @xmath593 , @xmath594 , and @xmath595 . as @xmath588 in @xmath284 , there must be another component @xmath596 of @xmath597 such that @xmath598 in @xmath284 . in particular , @xmath596 is essential in @xmath424 . choose an orientation of @xmath596 . then , @xmath596 has type @xmath599 where @xmath600 . the algebraic intersection number of @xmath134 and @xmath596 in @xmath424 equals @xmath601 , which must vanish since @xmath574 is embedded in @xmath424 . it follows that @xmath596 has type @xmath602 . switching the orientation of @xmath596 if necessary , @xmath596 has type @xmath581 . thus , @xmath134 and @xmath596 are parallel in @xmath424 . an exercise ( left to the reader ) shows that the linking number @xmath603 in @xmath0 equals @xmath604 . as @xmath134 and @xmath596 are disjointly embedded in the sphere @xmath370 , @xmath605 . hence , @xmath583 as desired . we define an * extended segment * to be an arc component of @xmath606 of the form : @xmath607{bar_arrow.eps } , \quad \includegraphics[scale=0.6]{scurve_arrow.eps } , \quad { \textnormal{or}}\quad \includegraphics[scale=0.6]{scurve_straddle_arrow.eps}\ ] ] in the latter type , the vertical line is a component of @xmath31 and the horizontal lines are arcs of @xmath437 not meeting special points . [ int_ext_seg ] any component of @xmath606 is an extended segment . let @xmath566 be a component of @xmath606 . by lemma  [ scin ] , no   appears . therefore , @xmath608 . hence , @xmath566 is an arc neatly embedded in @xmath408 . note that @xmath609 , and @xmath610 is disjoint from @xmath31 and from special points . by lemma  [ taco ] , no   appears . the result now follows from lemma  [ scin ] by considering the possible locations of points in @xmath610 . [ disjoint ] @xmath459 . suppose , by way of contradiction , that @xmath133 meets @xmath408 . then , there is a circle @xmath611 . by corollary  [ int_ext_seg ] , there exists a finite , disjoint collection @xmath612 , @xmath613 , of extended segments such that : @xmath614 orient all segments @xmath615 , @xmath616 , to point out from @xmath617}$ ] . note that this yields a coherent orientation of @xmath134 . a moment of reflection on the immersion @xmath35 ( cf . figure  [ immersion ] ) reveals that each @xmath618 winds around @xmath424 by : ( i ) @xmath198 revolutions in the @xmath576 direction , and ( ii ) @xmath619 revolutions in the @xmath575 direction where @xmath620 . hence , @xmath134 has type @xmath581 where @xmath621 and @xmath622 , which contradicts lemma  [ essential ] . this completes our proof that @xmath19 is irreducible and our proof of theorem  [ a_irred ] . this section proves that certain spheres in concatenations of borromean blocks may be ambiently isotoped into a single block summand , while fixing the tangle setwise . [ borr_prime ] let @xmath23 , @xmath623 , be borromean blocks where @xmath332 . consider the concatenation : @xmath624},\tau\right)}\ ] ] let @xmath50 be a @xmath17-sphere embedded in the interior of @xmath625}$ ] , transverse to @xmath31 , and meeting each component @xmath61 of @xmath31 at exactly one point @xmath626 for @xmath449 , @xmath17 , and @xmath27 . then , there is a neighborhood @xmath235 of @xmath627}$ ] in @xmath625}$ ] and an ambient isotopy @xmath175 , @xmath237 , of @xmath625}$ ] such that :    1 .   @xmath177 . @xmath238 for all @xmath176 . 3 .   @xmath175 fixes @xmath31 setwise for all @xmath237 . 4 .   @xmath628}}$ ] for some @xmath629 .    for the definition of _ borromean block _ , see section  [ borr_blocks ] .    without loss of generality , no point @xmath626 lies in a sphere @xmath630}$ ] where @xmath631 , and @xmath50 is transverse to these spheres . so , @xmath632}\right)}$ ] is a closed @xmath276-manifold . it suffices to improve @xmath50 , by appropriate ambient isotopies of @xmath625}$ ] , so that @xmath89 becomes empty . we will employ the following two operations . + * operation 1 . * suppose @xmath134 is a component of @xmath89 bounding a disk @xmath633 such that : ( i ) @xmath289 is disjoint from @xmath634 and ( ii ) @xmath289 is disjoint from @xmath31 . the circle @xmath134 lies in @xmath630}$ ] for some @xmath631 and bounds two @xmath17-disks , @xmath290 and @xmath383 , in @xmath630}$ ] . by lemma  [ disk_ess_sphere ] ( with @xmath635 ) , we have @xmath636 and @xmath291 is inessential in @xmath625}$ ] . so , @xmath637 ( since each @xmath61 meets @xmath290 at most once ) and there is a @xmath27-disk @xmath77 in @xmath625}$ ] bounded by @xmath291 . hence , @xmath31 is disjoint from @xmath77 and @xmath77 permits construction of an ambient isotopy , with support near @xmath77 , that carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath89 . + * operation 2 . * suppose @xmath134 is a component of @xmath89 bounding a disk @xmath633 such that : ( i ) @xmath289 is disjoint from @xmath634 and ( ii ) @xmath289 intersects @xmath31 at one point . for notational convenience , we assume @xmath638 . note that @xmath639 . the circle @xmath134 lies in @xmath630}$ ] for some @xmath631 . let @xmath640}$ ] be the @xmath17-disk whose boundary is @xmath134 and whose interior meets @xmath42 ( necessarily at one point , call it @xmath137 ) . now , @xmath289 lies in @xmath641}$ ] or in @xmath642}$ ] . without loss of generality , assume the latter . consider the block : @xmath643},r\right)}\ ] ] where @xmath8 has components : @xmath644 } { \textnormal { for $ i=1 $ , $ 2 $ , and $ 3$.}}\ ] ] let @xmath413 be the block obtained from @xmath284 by forgetting @xmath645 . then , @xmath646 since @xmath647 is a borromean block . hence , lemma  [ ball - arc_diffeo ] implies that @xmath291 bounds the @xmath27-disk @xmath648}$ ] , that @xmath649 , and there exists a diffeomorphism @xmath650 . the first two of these consequences imply that @xmath651 . forgetting @xmath652 instead of @xmath645 , we get @xmath653 . therefore , we have a diffeomorphism : @xmath654 this diffeomorphism permits construction of an ambient isotopy of @xmath625}$ ] that : ( i ) has support near @xmath77 , ( ii ) fixes @xmath43 and @xmath404 pointwise , ( iii ) fixes @xmath42 setwise , and ( iv ) carries @xmath289 past @xmath290 to a parallel copy of @xmath290 . thus , @xmath134 ( at least ) has been eliminated from @xmath89 . + observe that if @xmath655 , then operation 1 or 2 is applicable . indeed , let @xmath596 be a component of @xmath89 . one component , @xmath656 , of @xmath657 contains at most one of the points @xmath658 , @xmath659 , or @xmath660 . if @xmath656 contains components of @xmath89 , then operation 1 or 2 applies to any innermost component of @xmath89 in @xmath656 . if @xmath656 contains no component of @xmath89 , then operation 1 or 2 applies to @xmath596 itself . so , by finitely many applications of operations 1 and 2 , we get @xmath661 and the proof is complete .    [ fin_cor ] let @xmath23 , @xmath662 , and @xmath663 , @xmath664 , be irreducible borromean blocks . consider the concatenations : @xmath665},\tau\right)}\\      c:=&c_1 c_2 \cdots c_n = { \left(s^2{\left[1,n+1\right]},\sigma\right)}\end{aligned}\ ] ] suppose there is a diffeomorphism @xmath666 such that @xmath667 . then , @xmath668 and there is an isotopy of @xmath35 , sending @xmath31 to @xmath32 setwise at all times , to a diffeomorphism @xmath669 such that @xmath670 for each @xmath671 . assume @xmath672 ( otherwise , consider @xmath673 ) . if @xmath674 , then @xmath675 contradicts irreducibility of @xmath26 ( recall that each @xmath676 since each @xmath663 is a borromean block ) . so , @xmath677 and we are done . next , let @xmath678 . all isotopies of @xmath35 send @xmath31 to @xmath32 setwise at all times . all isotopies of @xmath134 are ambient and fix @xmath32 setwise at all times . by an abuse of notation , the corresponding improved @xmath35 will still be denoted by @xmath35 . by proposition  [ borr_prime ] , we can isotop @xmath35 such that @xmath679\right)}\subset{{\textnormal{int } } \ , c}_j$ ] for some @xmath680 . as @xmath663 is irreducible , we can isotop @xmath35 so that @xmath679\right)}$ ] is a level @xmath17-sphere in @xmath663 very close to one boundary sphere of @xmath663 , namely : ( i ) @xmath681 $ ] , or ( ii ) @xmath682$].case 1 . @xmath683 and ( i ) then , @xmath684 , a contradiction since @xmath26 is a borromean block.case 2 . @xmath683 and ( ii ) occurs , or @xmath685 and ( i ) occurs . then , we may further isotop @xmath35 so that @xmath679\right)}=s^2[2]$].case 3 . @xmath685 and ( ii ) occurs , or @xmath686 . these cases contradict irreducibility of @xmath26.in any case , we have arranged that @xmath679\right)}=s^2[2]$ ] . repeat this process with @xmath687\right)}$ ] and so forth , and it must terminate with @xmath668 . [ workhorse ] let @xmath23 and @xmath33 , @xmath148 , be sequences of irreducible borromean blocks . consider the two borromean rays : @xmath688 suppose there is a diffeomorphism @xmath34 . then , there exist @xmath689 and @xmath690 , and there is an isotopy of @xmath35 , sending @xmath31 to @xmath32 setwise at all times , to a diffeomorphism @xmath36 such that @xmath691 for each @xmath692 . all isotopies send @xmath31 to @xmath32 setwise at all times , and are relative to @xmath313 . by compactness , there exists an integer @xmath693 such that @xmath694})$ ] is disjoint from @xmath695 . by proposition  [ borr_prime ] , we may isotop @xmath35 so that @xmath696\right)}\subset { { \textnormal{int } } \ , c}_i$ ] for some @xmath697 . as @xmath33 is irreducible , we may further isotop @xmath35 so that @xmath696\right)}=s^2[m]$ ] for some integer @xmath698 . define @xmath699 . use proposition  [ borr_prime ] and irreducibility repeatedly to get @xmath700\right)}=s^2[i+n]$ ] for each integer @xmath692 . this section proves theorem  [ borr_rays_thm ] , the first of our main results . if @xmath32 is a multiray given by a concatenation of blocks : @xmath701 then the * mirror * of @xmath32 , denoted @xmath702 , is defined by : @xmath703    [ borr_rays_thm ] let @xmath23 and @xmath33 , @xmath24 , be sequences of blocks in @xmath18 . consider the two borromean rays : @xmath688 there exists a diffeomorphism @xmath34 preserving orientation of @xmath0 if and only if there exists @xmath689 such that one of the following holds for all sufficiently large @xmath24 :          if  [ beqc ] holds , then @xmath35 exists by remarks  [ concat_remarks ] item  [ tail_det_type ] . assume  [ beqcbarstar ] holds . then : @xmath711 the last diffeomorphism is @xmath712 (= rotation of @xmath0 about @xmath118-axis ) , followed by a simple ambient isotopy in each block relative to boundary @xmath17-spheres ( cf . section  [ s : diffeo_blocks ] ) . the other two diffeomorphisms come from remarks  [ concat_remarks ] item  [ tail_det_type ] . the composition is the required @xmath35 . + assume  [ beqcbar ] holds . the first orientation preserving case above yields a diffeomorphism @xmath713 that preserves orientation of @xmath0 . composing @xmath197 with @xmath714 yields the required @xmath35 . assume  [ beqcstar ] holds . then : @xmath715 for sufficiently large @xmath296 . the second orientation preserving case yields a diffeomorphism @xmath713 that preserves orientation of @xmath0 . again , compose @xmath197 with @xmath714 to obtain the required @xmath35 . + for the forward implications , note that blocks in @xmath18 are irreducible by theorem  [ a_irred ] . by corollary [ workhorse ] , there exist @xmath689 , @xmath690 , and a diffeomorphism @xmath36 , isotopic to @xmath35 , such that @xmath691 for each @xmath692 . assume @xmath35 preserves orientation of @xmath0 . then , @xmath197 preserves orientation of @xmath0 . so , each of the diffeomorphisms : @xmath716 preserves orientation , preserves boundary @xmath17-spheres setwise , and has the same tangle permutation @xmath717 ( see section  [ s : diffeo_blocks ] ) . if @xmath718 , then  [ beqc ] holds by table  [ hom_block ] in section  [ s : diffeo_blocks ] . if @xmath719 , then  [ beqcbarstar ] holds by table  [ hom_block ] . + finally , assume the given @xmath35 reverses orientation of @xmath0 . the composition of @xmath35 with @xmath714 is a diffeomorphism @xmath720 preserving orientation of @xmath0 . now , apply the orientation preserving case . let @xmath721 denote the set of all sequences @xmath23 , @xmath24 , of blocks in @xmath18 . declare two sequences to be equivalent , written @xmath722 , if and only if their corresponding borromean rays are equivalent by some diffeomorphism of @xmath0 ( not necessarily orientation preserving ) . a multiray @xmath724 is * achiral * provided there exists a diffeomorphism @xmath725 preserving orientation of @xmath0 . otherwise , @xmath31 is * chiral*. equivalently , @xmath31 is * achiral * provided there exists a diffeomorphism @xmath726 reversing orientation of @xmath0 .    [ chiral_cor ] let @xmath727 for a sequence @xmath23 , @xmath24 , of blocks in @xmath18 . then , @xmath31 is achiral if and only if ( i ) there exists a block @xmath728 where @xmath729 and each @xmath730 , and ( ii ) a tail of the sequence @xmath23 equals one of the following : @xmath731 in particular , if @xmath31 is achiral , then @xmath23 is eventually periodic . so , at most countably many achiral @xmath31 arise this way .    by theorem  [ borr_rays ] , @xmath31 is achiral if and only if ( i ) @xmath23 and @xmath732 have identical tails , or ( ii ) @xmath23 and @xmath733 have identical tails . so , if a tail of @xmath23 has the form  , then @xmath31 is achiral . conversely , suppose @xmath23 and @xmath732 have identical tails ( the other case is similar ) . then , there exists @xmath689 and @xmath734 such that @xmath735 as @xmath736 , we get @xmath737 . without loss of generality , assume @xmath738 ( otherwise , apply bar to  ) . note that : @xmath739 where the second equality used  . repeating this argument , we get that if @xmath740 and @xmath741 , then : @xmath742 therefore , @xmath743 . [ achiral_br ] for each @xmath744 , define : @xmath745 and define : @xmath746 by corollary  [ chiral_cor ] , @xmath747 is achiral . by theorem  [ borr_rays_thm ] , @xmath748 is a diffeomorphism invariant of @xmath747 . so , @xmath747 , @xmath744 , is a countably infinite family of achiral borromean rays , pairwise distinct up to diffeomorphism . a * hyperplane * is a smooth , proper embedding of @xmath4 in @xmath749 . a * multiple hyperplane * @xmath750 is a smooth , proper embedding of a disjoint union of ( at most countably many ) copies of @xmath4 . the basic invariant of @xmath751 is an associated tree @xmath752 . the vertices of @xmath752 are the components of @xmath753 . two vertices are adjacent provided their closures in @xmath1 share a component of @xmath751 . figure  [ hyp_graphs ] depicts some multiple hyperplanes and their trees . multiple hyperplanes in @xmath1 , @xmath6 , are classified by their associated trees  @xcite . more precisely , if @xmath751 and @xmath754 are multiple hyperplanes in @xmath1 , @xmath755 , and the trees @xmath752 and @xmath756 are isomorphic , then there is a diffeomorphism @xmath757 that preserves orientation of @xmath1 . for @xmath758 , these trees are naturally planar ( i.e. , the edges incident with a given vertex are cyclically ordered ) . the result then holds provided @xmath752 and @xmath756 are isomorphic as planar trees . + let @xmath759 denote klein s model of hyperbolic @xmath27-space . namely , @xmath759 is the open unit @xmath27-disk in @xmath0 , and a * hyperbolic hyperplane * is the nonempty intersection of @xmath759 with an affine plane in @xmath0 . a * hyperbolic multiple hyperplane * is a properly embedded submanifold of @xmath759 , each component of which is a hyperbolic hyperplane . the multiple hyperplanes in figure  [ hyp_graphs ] are hyperbolic . a multiple hyperplane @xmath760 is * unknotted * provided there exists a diffeomorphism @xmath761 where @xmath762 is a hyperbolic multiple hyperplane . let @xmath751 and @xmath754 be unknotted multiple hyperplanes in @xmath0 . there exists a diffeomorphism @xmath763 preserving orientation of @xmath0 if and only if @xmath752 and @xmath756 are isomorphic trees .      given a tree @xmath20 that is at most countable and is not necessarily locally finite , it is not difficult to construct a hyperbolic multiple hyperplane @xmath764 such that @xmath765 is isomorphic to @xmath20 . hence , unknotted multiple hyperplanes in @xmath0 are classified , up to diffeomorphism , by isomorphism classes of such trees . up to isomorphism , there is a unique tree with @xmath37 vertices for @xmath766 . for @xmath767 , there are two : a linear tree and the @xmath27-prong ( see figure  [ hyp_graphs ] ) . + let @xmath768 denote closed upper half space . throughout this section , @xmath769 denotes the ray @xmath770 in the positive @xmath120-axis . + if @xmath771 is a multiray with @xmath772 components , then @xmath773 denotes a smooth , closed regular neighborhood of @xmath8 in @xmath0  @xcite ( see also hirsch  @xcite ) . if @xmath183 is a component of @xmath8 , then @xmath774 denotes the component of @xmath773 containing @xmath183 . basic properties of @xmath773 include :    1 . [ mnr1 ] for each @xmath183 , there is a diffeomorphism of pairs @xmath775 . [ mnr2 ] @xmath773 is unique up to ambient isotopy of @xmath0 relative to @xmath8 . [ mnr3 ] the boundary of @xmath773 , denoted @xmath776 , is an @xmath37 component multiple hyperplane . [ mnr4 ] the tree @xmath777 is an @xmath37-prong . let @xmath30 be the ray @xmath783 in the positive @xmath118-axis . let @xmath784 denote the points of norm @xmath785 in the first quadrant of the @xmath579-plane . we identify @xmath784 with @xmath65\times[1,\infty)$ ] so that @xmath786 and @xmath787 . all isotopies of @xmath781 will be ambient and relative to a neighborhood of @xmath780 in @xmath781 . isotoped subsets will be called by their original names . it suffices to construct an isotopy of @xmath781 that carries @xmath788 to @xmath789 . let @xmath790 be orthogonal projection . the submanifold @xmath791 permits construction of an isotopy of @xmath781 that carries @xmath788 close enough to @xmath792 so that @xmath793 is an embedding . this is possible since @xmath792 is properly embedded . next , by integrating a suitable vector field of the form @xmath794 , we get an isotopy carrying @xmath788 into @xmath795 . rays do not knot in @xmath796  ( * ? ? ? .  9.13 ) . so , there is an ambient isotopy @xmath211 of @xmath795 that carries @xmath788 to a straight ray . use a small tube about @xmath795 in @xmath797 and a suitable bump function to extend @xmath211 to an isotopy of @xmath781 . the rest is elementary .    1 .   lemma  [ ztoz ] is very different from ` uniqueness of regular neighborhoods ' . let @xmath133 be a smooth subcomplex of a smooth manifold @xmath798 . it is not true , in general , that each orientation preserving diffeomorphism @xmath799 is isotopic to a diffeomorphism @xmath800 . for a simple counterexample , let @xmath133 be a bouquet of three circles embedded in @xmath801 in such a way that no circle of @xmath133 is inside another . then , @xmath802 is a smooth , compact @xmath17-disk with three holes . let @xmath803 be a simple closed curve such that two boundary components of @xmath804 are inside @xmath134 . let @xmath805 be a dehn twist about @xmath134 . then , @xmath220 is not isotopic to a diffeomorphism @xmath800 . otherwise , @xmath220 would be isotopic to the identity , which is false  @xcite . counterexamples exist even when @xmath133 is a smooth submanifold . we are indebted to bob gompf for these examples . let @xmath284 be a simply - connected , closed , symplectic @xmath806-manifold with positive signature , denoted @xmath807 , and @xmath808 . ( many such manifolds are known  even khler examples have been around for several decades . ) as @xmath284 is symplectic , @xmath284 is smooth and oriented , and @xmath809 is odd . let @xmath133 be @xmath284 blown up @xmath810 times . then , @xmath811 and @xmath133 is homeomorphic to @xmath812 for some @xmath678 ( and very large in practice ) . since @xmath133 is symplectic , @xmath133 has nonvanishing seiberg - witten invariants , denoted @xmath813 . however , @xmath814 (= @xmath133 with reversed orientation ) splits off a @xmath815 summand ( from the blowup of @xmath284 ) and has @xmath808 . therefore , @xmath816 and @xmath133 admits no orientation reversing self diffeomorphism . fix any @xmath817 . then , @xmath818 is a smooth , closed , oriented manifold containing @xmath819 , and @xmath820 is identified with @xmath821 . now , there is a smooth @xmath220-cobordism @xmath656 between @xmath133 and @xmath89 . so , @xmath822 is smoothly a product , and , working relative to boundary , @xmath823 is smoothly a product . hence , there is a diffeomorphism @xmath824 . let @xmath135 be an orientation reversing self diffeomorphism of @xmath89 ( e.g. , @xmath135 permutes the summands and is otherwise the identity ) . let @xmath136 be an orientation reversing self diffeomorphism of @xmath92 . then , @xmath825 is an orientation preserving self diffeomorphism of @xmath826 , and @xmath827 is multiplication by @xmath828 on @xmath829 . let @xmath220 be the corresponding orientation preserving self diffeomorphism of @xmath804 . on @xmath830 , @xmath831 is multiplication by @xmath828 . so , @xmath220 is not isotopic to a diffeomorphism @xmath800 . otherwise , we get a forbidden orientation reversing self diffeomorphism of @xmath133 . an alternative approach to proving lemma  [ ztoz ] uses the following lemma together with some collaring arguments . only two implications require proof . + [ ediffeo ] @xmath842 [ eisotopy ] : we may assume @xmath35 preserves orientation of @xmath1 ( otherwise , compose @xmath35 with reflection through @xmath843 ) . by milnor @xcite , @xmath35 is isotopic to the identity . + [ ehalves ] @xmath842 [ ediffeo ] : let @xmath844 denote closed lower half space . replacing @xmath845 and @xmath846 with their compositions with appropriate reflections , we can and do assume @xmath840 and @xmath847 are orientation preserving diffeomorphisms . let @xmath848 , an orientation preserving automorphism of @xmath843 . let @xmath849 be the orientation preserving automorphism of @xmath4 given by the following composition where @xmath850 is the obvious projection : @xmath851 let @xmath852 be the identity map on @xmath2 . then , @xmath853 is an orientation preserving automorphism of @xmath854 . replacing @xmath845 with @xmath855 , we can and do further assume @xmath856 . define @xmath857 by @xmath858 if @xmath859 and @xmath860 if @xmath861 . then , @xmath220 is an orientation preserving autohomeomorphism of @xmath1 . by construction , @xmath220 is smooth on @xmath284 and @xmath220 is smooth on @xmath836 . using collaring uniqueness  ( * ? ? ? 8.1.9 ) , we may adjust @xmath220 ( by isotoping @xmath845 and @xmath846 near @xmath751 and relative to @xmath751 ) to obtain the desired diffeomorphism @xmath35 . [ rtor ] let @xmath8 and @xmath185 be multirays in @xmath0 . let @xmath862 be a diffeomorphism . let @xmath778 and @xmath863 . then , @xmath220 is isotopic , relative to a neighborhood of @xmath751 in @xmath773 , to a diffeomorphism of pairs @xmath864 . it suffices to consider @xmath865 where @xmath183 is a component of @xmath8 . reindex the components of @xmath185 so that @xmath866 . let @xmath775 and @xmath867 be diffeomorphisms . lemma  [ ztoz ] yields an isotopy , relative to a neighborhood of @xmath780 , such that @xmath868 and @xmath869 . then , @xmath870 is the desired isotopy of @xmath865 . suppose @xmath751 is unknotted . then , there is a diffeomorphism @xmath872 where @xmath762 is a hyperbolic multiple hyperplane . as @xmath752 is an @xmath37-prong , so is @xmath765 . without loss of generality , the origin of @xmath759 does not lie in @xmath873 . for each component @xmath183 of @xmath8 , let @xmath874 and let @xmath875 . let @xmath70 be the point of @xmath876 closest to the origin in @xmath759 ( for the euclidean metric ) . let @xmath877 be the radial ray in @xmath759 with initial point @xmath70 . let @xmath878 be the radial ray in @xmath879 that is half as long as @xmath879 ( for the euclidean metric ) . notice that @xmath33 is a smooth , closed regular neighborhood of @xmath61 in @xmath759 . let @xmath880 be the radial multiray with components @xmath61 . notice that @xmath873 is a smooth , closed regular neighborhood of @xmath31 in @xmath759 . as in corollary [ rtor ] , we may isotop @xmath220 to a diffeomorphism @xmath881 . but , this implies @xmath8 is unknotted , a contradiction . [ rays_planes ] let @xmath31 and @xmath269 be multirays in @xmath0 , each containing @xmath37 components where @xmath882 . let @xmath883 and let @xmath884 . if @xmath885 is a diffeomorphism , then @xmath35 is isotopic relative to @xmath31 to a diffeomorphism @xmath886 . conversely , suppose @xmath886 is a diffeomorphism . if any of the following conditions are met , then @xmath197 is isotopic relative to @xmath887 to a diffeomorphism @xmath885 . the case @xmath677 where @xmath31 and @xmath269 are unknotted is exceptional for trivial reasons . for example , let @xmath31 and @xmath269 both equal @xmath789 . let @xmath889 and @xmath890 both equal @xmath781 , so @xmath891 . then , @xmath892 is an orientation preserving diffeomorphism of @xmath0 sending @xmath751 to @xmath754 . however , @xmath197 is not isotopic relative to @xmath751 to a diffeomorphism @xmath893 . the forward implication is immediate by ambient uniqueness of closed regular neighborhoods  @xcite . next , let @xmath886 be a diffeomorphism . first , assume condition  [ cond1 ] . then , the result is immediate by corollary  [ rtor ] . second , assume condition  [ cond2 ] . as @xmath752 and @xmath756 are both @xmath37-prongs , condition  [ cond1 ] is satisfied and the result follows . third , assume condition  [ cond3 ] . without loss of generality , assume @xmath31 is knotted ( otherwise , consider @xmath260 ) . let @xmath894 and let @xmath895 . by corollary  [ krikh ] , @xmath751 is knotted . by lemma [ unknot_equiv ] , @xmath896 . as @xmath897 , we see that condition  [ cond1 ] is satisfied and the result follows . proposition  [ rays_planes ] permits us to translate results on knotted multirays in @xmath0 to results on knotted multiple hyperplanes in @xmath0 . by mcpherson  @xcite , there exist uncountably many knot types of a ray : so there exist uncountably many knot types of a hyperplane . by fox and harrold @xcite , there exist uncountably many knot types of two component multirays with unknotted components ( see fox and artin  @xcite for a nice example ) : so there exist uncountably many knot types of two component multiple hyperplanes with unknotted components . + proposition  [ rays_planes ] is proved via ambient isotopies , so it also yields results on chirality . let @xmath898 be a multiple hyperplane . we say @xmath751 is * achiral * provided there is a diffeomorphism @xmath899 that reverses orientation of @xmath0 . otherwise , @xmath751 is * chiral*. for example , it is an exercise to show that each unknotted multiple hyperplane @xmath760 is achiral . + a multiple hyperplane @xmath751 in @xmath0 forms * borromean hyperplanes * provided @xmath751 is knotted , but any two components of @xmath751 form an unknotted multiple hyperplane . proposition  [ rays_planes ] implies that if @xmath30 forms borromean rays , then @xmath776 forms borromean hyperplanes . thus , we obtain our second main result . there exist uncountably many borromean hyperplanes , pairwise distinct up to diffeomorphism of @xmath0 . there exists a countably infinite family of achiral borromean hyperplanes , pairwise distinct up to diffeomorphism of @xmath0 . calcut , h.c .  king , and l.c . siebenmann , _ connected sum at infinity and cantrell - stallings hyperplane unknotting _ , accepted at rocky mountain j. math . ( 2011 ) , available at http://www.oberlin.edu/faculty/jcalcut/csi.pdf[http://www.oberlin.edu/faculty/jcalcut/csi.pdf ] , 42 pp .
three disjoint rays in @xmath0 form _ borromean rays _ provided their union is knotted , but the union of any two components is unknotted . 
 we construct infinitely many borromean rays , uncountably many of which are pairwise inequivalent . 
 we obtain uncountably many borromean hyperplanes .