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the periodic orbits of a flow in @xmath0 form knots . for _ morse - smale flows _ there are only finitely many periodic orbits . wada @xcite has classified all links that can be realized as a collection of closed orbits of nonsingular morse - smale flows on @xmath0 . further , wada s scheme includes an indexing of the components of the link according to whether the orbit is an attractor , a repeller , or a saddle .    in a _ smale flow _ , by contrast , the saddle sets may contain infinitely many closed orbits , while the attractors and repellers must still be collections of finitely many orbits . franks @xcite devised an abstract classification scheme for nonsingular smale flows on @xmath0 using a device he called the _ lyapunov graph _ of a flow . each vertex of a lyapunov graph corresponds to an attractor , repeller or saddle set ( the _ basic sets _ of the flow ) . the saddle vertices are labeled with an _ incidence matrix _ ( determined non - uniquely by the first return map on a cross section ) . a simple algorithm is used to decide if a given lyapunov graph can be realized by a nonsingular smale flow on @xmath0 . however , the lyapunov graph contains no explicit information about the embedding of the basic sets . in contrast to wada s study of morse - smale flows , franks work does does not allow one to _ see _ smale flows . it was our curiosity to visualize smale flows that motivated this paper . it is however worth noting that wada s results have provided tools for understanding bifurcations between morse - smale flows @xcite and it is likely that some of our results may sheld light on bifurcations between smale flows and form morse - smale flows to smale flows . also see @xcite for an example . the project of this paper is to visually construct examples of smale flows and in some special contexts classify all the possible embedding types . our primary tools will be the theory of _ templates _ , branched 2-manifolds which model the saddle sets @xcite and certain earlier results of franks that do give some information about the embedding of closed orbits . specifically , computations of linking numbers and alexander polynomials are employed . sections [ sec1 ] and [ sec2 ] contain background information . our main classification theorem ( theorem  [ thm_ls ] ) is in section [ sec3 ] . various generalizations and applications follow in sections [ sec4 ] and [ sec5 ] . the author wishes to thank john franks and masahico saito for helpful conversations . a knot @xmath1 is an embedding of @xmath2 into @xmath0 . it is traditional to use @xmath1 to denote both the embedding function and the image in @xmath0 . a knot may be given an orientation . we will always use a flow to induce an orientation on our knots . the _ knot group _ of @xmath1 is the fundamental group of @xmath3 . a link of @xmath4 components is an embedding of @xmath4 disjoint copies of @xmath2 into @xmath0 . two knots @xmath5 and @xmath6 ( or two links ) are equivalent if there is an isotopy of @xmath0 that takes @xmath5 to @xmath6 . when we talk about a knot we almost always mean its equivalence class , or _ knot type_. a _ knot diagram _ is a projection of a knot or link into a plane such that any crossings are transverse . if the knot has been given an orientation the crossings are then labeled as positive or negative according to whether they are left - handed or right - handed respectively . the knot group can be calculated from a diagram and , unlike the diagram , is invariant . if a knot has a diagram with no crossings then it is called an _ unknot _ or a _ trivial knot_. the following proposition will be of use to us .    * ( the unknotting theorem , @xcite ) * [ prop_unknotting ] the knot group of @xmath1 is infinite cyclic if and only if @xmath1 is the unknot .    given a diagram of a two component link @xmath7 the _ linking number _ of @xmath5 with @xmath6 is half the sum of the signs of each crossing of @xmath5 under @xmath6 and is denoted @xmath8 . the linking number is a link invariant , and thus is independent of the choice of the diagram . the hopf link shown in figure  [ fig_hopf ] . its linking number is @xmath9 , depending on the choice of orientations . a knot @xmath10 is _ composite _ if there exists a smooth 2-sphere @xmath11 such that @xmath12 is just two points @xmath13 and @xmath14 , and if @xmath15 is any arc on @xmath11 joining @xmath13 to @xmath14 then the knots @xmath16 @xmath17 are each nontrivial , ( i.e. not the unknot ) . we call @xmath5 and @xmath6 factors of @xmath1 and write @xmath18 of course the designation of the two components of @xmath19 as `` inside '' and `` outside '' is arbitrary . this implies @xmath20 . we call @xmath1 the connected sum of @xmath5 and @xmath6 . if a nontrivial knot is not composite then it is _ prime_. figure  [ prime ] gives an example . it shows how to factor the _ square knot _ into two trefoils . trefoils are prime . it was shown by schubert ( * ? ? ? * chapter 5 ) that any knot can be factored uniquely into primes , up to order . the unknot serves as a unit .    a knot which fits on a torus and wraps about it @xmath13 times longitudinally and @xmath14 times meridianally ( @xmath13 and @xmath14 must be relatively prime ) , is called a @xmath21 _ torus knot_. if @xmath1 is any knot , then a @xmath21 _ cable _ about @xmath1 , where @xmath13 and @xmath14 are relatively prime , is defined as follows . let @xmath22 be a solid torus neighborhood of @xmath1 , whose core is @xmath1 . let @xmath23 be a _ standard longitude _ of @xmath24 for @xmath1 , i.e. @xmath25 . now consider a torus @xmath26 with a @xmath21 knot on it . let @xmath27 be a homeomorphism that takes a standard longitude of @xmath26 to @xmath23 . the image of @xmath21 under this map is said to be a @xmath21 cable of @xmath1 or , @xmath28 . we shall extend the usual cabling notation a bit . let @xmath29 be a meridian of @xmath1 , @xmath30 be a standard longitude of @xmath1 , and @xmath31 be a loop bounding a disk on @xmath26 . ( a curve on a surface is _ inessential _ if it bounds a disk in the surface , and is _ essential _ if it does not . ) thus , we have tools with which to build up new knots from old ones . in wada s paper and in sections 4 and 5 here , one builds new flows from old ones using processes based in part on taking connected sums and forming cables . the cabling construction has been generalized in two ways . a knot @xmath32 is a _ satellite _ of a given knot @xmath1 if @xmath32 lives inside a tubular neighborhood @xmath1 and meets every meridianal disk . if the orientation of @xmath32 is always roughly the same as that of @xmath1 ( i.e. there is a fibration of @xmath22 by meridianal disks which are always transverse to @xmath32 ) , we say that @xmath32 is a _ generalized cable _ of @xmath1 . it is known that the satellite of a nontrivial knot is nontrivial @xcite , a fact which we shall make use of in case 3 in the proof of theorem  [ thm_ls ] . knot polynomials form an important class of knot and link invariants . we shall make use of the first known knot polynomial , the alexander polynomial . it can be readily calculated from either a knot diagram or the fundamental group . the latter approach will be of special importance to us . the reader who wishes to check our polynomial calculations should be able to find all he or she needs in @xcite . a @xmath33 flow @xmath34 on a compact manifold @xmath35 is called _ structurally stable _ if any sufficiently close approximation @xmath36 in the @xmath33 topology is _ topologically equivalent _ , that is if there exists a homeomorphism @xmath37 taking orbits of @xmath34 to orbits of @xmath36 , preserving the flow direction . structurally stable @xmath33 flows have a hyperbolic structure on their chain - recurrent sets @xcite . we define these concepts next . a point @xmath38 is _ chain - recurrent _ for @xmath34 if for every @xmath39 and @xmath40 there exists a chain of points @xmath41 in @xmath35 , and real numbers @xmath42 all bigger than @xmath26 such that @xmath43 when ever @xmath44 . the set of all such points is called the chain - recurrent set @xmath45 . it is a compact set invariant under the flow . a compact invariant set @xmath46 for a flow @xmath34 has a _ hyperbolic structure _ if the tangent bundle of @xmath46 is the whitney sum of three bundles @xmath47 , @xmath48 , and @xmath49 each of which invariant under @xmath50 for all @xmath51 . furthermore , the vector field tangent to @xmath34 spans @xmath49 and there exist real numbers @xmath52 and @xmath53 such that @xmath54 @xmath55    we also define the local stable and unstable manifolds associated to an orbit @xmath56 . they are respectively , @xmath57 and @xmath58 the global stable and unstable manifolds are defined similarly by removing the condition that @xmath59 . it was shown by smale that if the chain - recurrent set @xmath45 of flow has a hyperbolic structure then @xmath45 is the union of a finite collection of disjoint invariant compact sets called the _ basic sets_. each basic set @xmath60 contains an orbit whose closure contains @xmath60 . the periodic orbits of a basic set @xmath60 are known to be dense in @xmath60 . a basic may either consist of a single closed orbit or it may contain infinitely many closed orbits and infinitely other nonperiodic chain - recurrent orbits . in the later case any cross section is a cantor set and the first return map is a _ subshift of finite type_. in the former case any cross section is a finite number of points but the first return map is still a ( trivial ) subshift of finite type . thus , each basic set is a suspension of a subshift of finite type . a nontrivial basic will be called _ chaotic_.    a flow @xmath61 on a manifold @xmath35 is called a _ smale flow _ provided    * the chain - recurrent set @xmath45 of @xmath61 has a hyperbolic structure , * the basic sets of @xmath45 are one - dimensional , and * the stable manifold of any orbit in @xmath45 has transversal intersection with the unstable manifold of any other orbit of @xmath45 . most references allow for fixed points but we will be working primarily with nonsingular flows . smale flows on compact manifolds are structurely stable under @xmath33 perturbations but are not dense in the space of @xmath33 flows . it is easy to see that for dim @xmath35 = 3 , each attracting and repelling basic set is a closed orbit . the admissible saddle sets , however , may be chaotic . a smale flow with no chaotic saddle sets is called a _ morse - smale _ flow .    for a chaotic saddle set of a smale flow in a 3-manifold one can construct a neighborhood that is foliated by local stable manifolds of orbits in the flow . collapsing in the stable direction produces a branched 2-manifold . with a semi - flow induced from the original flow , this branched 2-manifold becomes what is known as a _ template_. the template models the basic saddle set in that the saddle set itself can be recovered from the template via an inverse limit process and that any knot or link of closed orbits in the flow is smoothly isotopic to an equivalent knot or link of closed orbits in the template s semi - flow . the proof of this is due to birman and williams @xcite and can also be found in ( * ? ? ? * theorem 2.2.4 ) . a key tool in the analysis of hyperbolic flows is the concept of a markov partition . we refer the reader to @xcite for details . in our context a markov partition is a finite , disjoint collection of disks transverse to a basic set of a flow . each orbit of the basic set must pass through some element of the markov partition in forward time . given a markov partition @xmath62 for a suspended subshift of finite type with first return map @xmath63 we define the corresponding @xmath64 _ incidence matrix _ @xmath65 , by @xmath66    the incidence matrix , like a knot diagram , is not invariant but does contain invariant information . we can encode additional information about the embedding of a basic set by modifying the incidence matrix .    given a markov partition for a basic set with first return map @xmath63 , assign an orientation to each partition element . if the partition is fine enough the function @xmath67 is constant on each partition element . the _ structure matrix _ @xmath68 is then defined by @xmath69 , where @xmath70 is any point in the @xmath71-th partition element . the next proposition was proved by franks in @xcite . [ prop_link ] let @xmath61 be a smale flow with a single attracting closed orbit @xmath72 , and a single repelling closed orbit @xmath73 , with saddle sets @xmath74 . then the absolute value of the linking number of @xmath72 and @xmath73 is given by @xmath75 where @xmath76 denote the respective structure matrices of the saddle sets . the manner in which a saddle set `` links '' a closed orbit @xmath1 , is described by modifying the structure matrix @xmath68 to form a _ linking matrix _ @xmath77 . consider a markov partition , @xmath62 , of the saddle with incidence matrix @xmath65 . pick a base point @xmath78 in @xmath79 and paths @xmath80 from @xmath78 to @xmath81 , also in @xmath79 . for each @xmath82 let @xmath83 be a segment of the flow going from @xmath81 to @xmath84 without meeting any of the other partition elements . now form a loop consisting of @xmath83 , @xmath80 , @xmath85 and , if needed , short segments in @xmath81 and in @xmath84 . if the disks have been chosen small enough , then the linking number of any such loop with @xmath1 depends only on @xmath71 and @xmath86 . that this can always be done is shown in @xcite . the _ linking matrix _ @xmath87 associated with a suitable markov partition for a given closed orbit @xmath1 is defined to by @xmath88 where @xmath14 is the linking number of the loops formed from segments connecting @xmath81 to @xmath84 and @xmath1 as described above . the following proposition is a special case of theorem 4.1 in @xcite . [ franks , 1981 ] [ prop_alex ] let @xmath61 be a smale flow in @xmath0 with one attracting closed orbit @xmath72 , one repelling closed orbit @xmath73 , and a single saddle set @xmath89 . let @xmath90 and @xmath91 be the linking matrices for @xmath89 with respect to @xmath72 and @xmath73 respectively . then the alexander polynomials of @xmath72 and @xmath73 are given by @xmath92 and @xmath93 respectively , up to multiples of @xmath94 .    finally , we record a proposition about morse - smale flows . it is an obvious corollary to wada s theorem @xcite , though it is easy to prove directly . [ prop_hopf ] in a ( nonsingular ) morse - smale flow on @xmath0 with exactly two closed orbits , the link of closed orbits forms a hopf link with one component a repeller and the other an attractor . by a _ simple smale flow _ we shall mean a smale flow with three basic sets : a repelling orbit @xmath73 , an attracting orbit @xmath72 , and a nontrivial saddle set . in this section we will show how to construct simple smale flows in which the saddle set @xmath23 can be modeled by an embedding of the lorenz template shown in figure  [ lorenz ] . that is , there is an isolating neighborhood of the saddle set @xmath23 foliated by local stable manifolds of the flow , such that when we collapse out in the stable direction we get an embedding of the lorenz template . call such flows _ lorenz - smale flows_. in this section we classify all possible lorenz - smale flows . we shall call the isolating neighborhood of the saddle set @xmath87 . the set of points of @xmath95 where the flow is transverse outward is called the _ exit set_. the backward orbits of these points approach orbits in the saddle set @xmath23 . the exit set consists of two annuli , @xmath96 and @xmath97 , connected by a long strip @xmath68 . the core of the exit set is homeomorphic to the boundary of the lorenz template . we shall refer to the cores of @xmath96 and @xmath97 , as @xmath70 and @xmath98 , respectively . see figure  [ lnbhd ] . the set of points of @xmath95 where the flow is transverse inward is called the _ entrance set_. the entrance set also consists of two annuli connected by a long strip . denote these two annuli by @xmath99 and @xmath100 and their respective cores @xmath101 and @xmath102 and the connecting strip by @xmath103 . see figure  [ lnbhd ] . the intersection of the closures of the entrance and exit sets consists of three closed curves where the flow is tangential to @xmath95 . although the entrance set is harder to visualize , its topological type can be determined by an euler characteristic argument . notice that @xmath70 and @xmath101 are isotopic to @xmath104 and hence have the same knot type . similarly , @xmath98 and @xmath102 must have the same knot type .    take tubular neighborhoods of @xmath72 and @xmath73 and denote them by @xmath65 and @xmath105 respectively .    to build a smale flow from these building blocks , we first attach the closure of the exit set of @xmath87 to @xmath106 . this gives a vector field on a new 3-manifold pointing inward along its entire boundary . that this can be done smoothly was shown by morgan in @xcite . next attach @xmath107 to the boundary of @xmath108 so that the union @xmath109 is @xmath0 . we know from proposition  [ prop_link ] that the linking number between @xmath73 and @xmath72 must be @xmath9 , but what types of knots can @xmath72 and @xmath73 be ? what are all the different ways the saddle set can be embedded so as to still have the lorenz template as a model ? this later question is made precise by asking , what types of knots can @xmath70 and @xmath98 be ? can they be linked ? can the annuli @xmath96 and @xmath97 have any number of twists or are there restrictions ? ( this last question is equivalent to finding the _ framing _ of a knot . )    to answer these questions we use the following framework . first , we study what may happen when the attaching of the exit set of @xmath87 to @xmath106 is such that @xmath70 and @xmath98 are both inessential in @xmath106 . then we investigate the case where one of them , say @xmath98 , is essential but the other is still inessential . finally we consider the case where both @xmath70 and @xmath98 are essential in @xmath106 . the results of this analysis are then consolidated into the statement of theorem  [ thm_ls ] . our classification scheme is only up to isotopy of @xmath0 , plus mirror images and flow reversal . also , we shall not be concerned with the orientation , i.e. the flow direction , of @xmath72 or @xmath73 , since their orientations can be easily reversed by modifying the flow in a tubular neighborhoods of @xmath72 and @xmath73 , ( * ? ? ? * first paragraph ) . [ thm_ls ] for a lorenz - smale flow in @xmath0 the following and only the following configurations are realizable . the link @xmath110 is either a hopf link or a trefoil and meridian . in the latter case the saddle set is modeled by a standardly embedded lorenz template , i.e. both bands are unknotted , untwisted , and unlinked , with the core of each band a meridian of the trefoil component of @xmath110 . in the former case there are three possibilities : ( 1 ) the saddle set is standardly embedded . ( 2 ) one band is twisted with @xmath4 full - twists for any @xmath4 , but remains unknotted and unlinked to the other band , which must be unknotted and untwisted . ( 3 ) one band is a @xmath21torus knot , for any pair of coprime integers , with twist @xmath111 . the other band is unknotted , untwisted and unlinked to the knotted one . the proof is divided into three cases .    * case 1 : * suppose both @xmath70 and @xmath98 are inessential in @xmath106 . it follows that @xmath96 and @xmath97 are untwisted , that is the linking number between each of the two components of @xmath112 and of @xmath113 is zero . it is also obvious that @xmath70 and @xmath98 are unknotted and unlinked . there are two subcases to consider . it could be that @xmath70 and @xmath98 are concentric in @xmath106 , or it could be that they are not . that is @xmath70 and @xmath98 may or may not form the boundary of an annulus in @xmath106 . in figure  [ lonaball ] we show that both cases can be realized . the neighborhood @xmath87 is attached to a 3-ball @xmath114 along the closure of the exit set of @xmath87 . figure  [ lonaball ] also shows two ways one might attach handles to the 3-ball so as to turn it into a solid torus . suppose we attach the handle to the small disks marked @xmath115 and @xmath116 in the manner shown . call the resulting solid torus @xmath117 . if we take @xmath118 the result is still a solid torus , and the complement in @xmath0 is just another solid torus , @xmath119 . we can now build a smale flow with an attractor in @xmath117 , a repeller in @xmath119 and a lorenz saddle set in @xmath87 . upon further inspection the reader should be able to see that @xmath70 and @xmath98 are concentric . now , instead on attaching a handle at @xmath115 and @xmath116 , we attach one to @xmath114 and @xmath120 as shown again in figure  [ lonaball ] . this time call the solid torus obtained @xmath121 . as before @xmath122 is a solid torus with solid torus complement in @xmath0 . thus , we have constructed another lorenz - smale flow with @xmath70 and @xmath98 inessential on a tubular neighborhood of the attractor . is it diffeomorphic to the lorenz - smale flow we constructed before ? to see that the answer is no , study the loops @xmath70 and @xmath98 again . they are still both inessential , that is they both bound disks in @xmath123 . but they are no longer concentric . this can be seen from careful study of figure  [ lonaball ] .    in both these examples the attractor and repeller form a hopf link . we claim that if both @xmath70 and @xmath98 are inessential in @xmath106 then @xmath72 and @xmath73 must form a hopf link . since @xmath105 is a tubular neighborhood of a knot , the union of @xmath65 and @xmath87 is a knot complement . but we will show that @xmath108 is a solid torus whose core has the same knot type as that of @xmath72 . thus , we could remove @xmath108 from our flow and replace it with a solid torus containing just an attractor and no saddle set . this gives us a nonsingular morse - smale flow on @xmath0 with just two closed orbits whose link type is the same as @xmath110 . but by proposition  [ prop_hopf ] these must form a hopf link . we now show that @xmath108 is a solid torus of whose core is the same knot type as @xmath72 . first , assume that @xmath70 and @xmath98 are not concentric . then they bound disjoint disks @xmath124 and @xmath125 in @xmath106 . expand @xmath124 and @xmath125 , if needed , so that they contain all of @xmath96 and @xmath97 respectively but remain disjoint . thicken @xmath124 and @xmath125 by pushing into @xmath65 a little , forming two 3-balls @xmath126 and @xmath127 , which are disjoint and do not meet the orbit @xmath72 . let @xmath128 and @xmath129 . it is clear that @xmath130 is a solid torus whose core is still @xmath72 . the set @xmath131 is a 3-ball and @xmath132 . but the union @xmath133 is taken along the disk @xmath134 . thus , there is a deformation retract from @xmath135 to @xmath130 . this proves our claim for @xmath70 and @xmath98 not concentric . now suppose @xmath70 and @xmath98 are concentric and assume @xmath70 is inner most . then construct the 3-ball @xmath126 as before . let @xmath136 and @xmath137 . this time @xmath131 is a solid torus . it is attached to @xmath130 along the annulus @xmath138 whose core @xmath98 is inessential in @xmath139 and is a ( 1,0 ) longitude in @xmath131 . thus we can retract @xmath131 to @xmath97 and push it into @xmath130 without changing the knot type of @xmath72 . this finishes the proof of our claim . * case 2 : * suppose that @xmath70 is inessential but that @xmath98 is essential . the opposite case is similar . again , it is clear that @xmath96 is untwisted and that @xmath70 is unknotted and unlinked to @xmath98 . we shall again show that @xmath72 and @xmath73 must form a hopf link . it then follows that since @xmath98 lives in a standardly embedded torus , @xmath106 , @xmath98 is a torus knot or unknot . if @xmath98 is a meridian ( 0,1 ) , or a longitude ( 1,0 ) then @xmath97 is untwisted . if @xmath98 is an unknot @xmath140 or @xmath141 then @xmath97 has @xmath14 full twists . for nontrivial torus knots the twisting in @xmath97 is uniquely determined by the knot type of @xmath98 . if @xmath98 is a @xmath21 torus knot then the twist in @xmath97 is @xmath111 .    that any @xmath21 torus knot can be realized by @xmath98 is shown by construction in figure  [ yess ] . one places a @xmath21 curve on a torus . attach an annulus to this curve along one boundary component . add a `` lorenz ear '' to form a lorenz template . next a `` finger '' pushes out of the torus and snakes along the boundary of the template and finally pokes through the @xmath70 loop . thicken this complex up to get @xmath142 . the repeller is then placed as a meridian in the complement . an example with @xmath98 a @xmath143 curve is shown in figure  [ yessex ] .    the argument that @xmath72 and @xmath73 must form a hopf link is the same as in the concentric subcase of case 1 above . the core @xmath98 of the annulus @xmath97 is a @xmath140 cabling of the core of the solid torus @xmath131 . we can foliate @xmath131 with meridianal disks each of which meets @xmath97 in an arc . thus , @xmath97 is a deformation retraction of @xmath131 . we then push @xmath97 into @xmath130 . * case 3 : * suppose that both @xmath70 and @xmath98 are essential in @xmath106 . in figure  [ landt ] we give an example . the loops @xmath70 and @xmath98 are meridians in @xmath106 . they are also standard longitudes in @xmath107 . the attractor @xmath72 is a trefoil knot while @xmath73 is unknotted and is a meridian of @xmath72 . we claim that up to mirror images and flow reversal , this is the only possible configuration . now consider the general setting . the attaching map from @xmath95 to @xmath106 takes @xmath70 and @xmath98 to two copies of some @xmath21 cable knot of the attractor , @xmath72 . here we allow @xmath13 or @xmath14 to be zero , but not both . likewise , the attaching map from @xmath95 to @xmath107 takes @xmath101 and @xmath102 to some pair of @xmath144 knots . of course @xmath70 and @xmath98 are respectively ambient isotopic to @xmath101 and @xmath102 within @xmath95 , so all four have the same knot type .    for future reference , let @xmath145 . a knot @xmath1 with @xmath146 is a left or right handed trefoil . see ( * ? ? ? * lemma 15.37 , corollary 15.23 ) . the alexander polynomials of the attractor and repeller are @xmath147 and @xmath148 , respectively    if the linking matrix for @xmath72 is @xmath149,\ ] ] then the result follows by proposition  [ prop_alex ] . the only difficulty in determining the linking matrix is the assignment of the signs to the powers of the @xmath51 s . one can check our assignment explicitly for the @xmath150 case by studying figure  [ landt ] . in general , if the powers are all of the same sign , the polynomial that results is not symmetric in @xmath51 , nor is any @xmath94 multiple . but it is well known that the alexander polynomial of a knot is symmetric in @xmath51 , up to multiples of @xmath94 . that is @xmath151 see @xcite . @xmath152    not ready :    [ lem_11 ] the fundamental groups of @xmath153 and @xmath154 are @xmath155 and @xmath156 , respectively . the alexander polynomials of @xmath72 and @xmath73 are @xmath157 and @xmath158 , respectively . we shall find @xmath159 using the seifert / van kampen theorem . the calculations of @xmath160 are similar . the alexander polynomials can then be determined from fox s free differential calculus @xcite . we must choose generators for @xmath87 , @xmath105 and @xmath161 . the generators for @xmath161 and @xmath87 are shown in figure  [ generators ] . the base point @xmath162 is in the `` middle '' of the strip @xmath103 . for @xmath87 we shall abuse notation slightly and call the generators @xmath70 and @xmath98 , as they are isotopic to the @xmath70 and @xmath98 loops , however , we do not use the orientation of the flow . ( by the proof of the previous lemma the images of @xmath70 and @xmath98 must wrap around @xmath107 in the same direction . ) denote the generators of @xmath161 by @xmath163 and @xmath164 . for @xmath105 we shall use a loop isotopic to @xmath73 but with base point @xmath165 . again we abuse notation and call this new loop @xmath73 . the fundamental groups of interest are then @xmath166 @xmath167 and @xmath168 the homomorphisms induced by inclusion maps are @xmath169 and @xmath170 . these give @xmath171 @xmath172 @xmath173 and @xmath174 . by van kampen s theorem @xmath175 . @xmath152    it follows from lemmas 10 and 11 that if @xmath70 is a @xmath21 curve on @xmath106 then @xmath101 is a @xmath176 curve on @xmath107 . if @xmath13 or @xmath14 is zero then the other is @xmath9 since the curve is in a torus . now suppose @xmath70 is a @xmath177 curve on @xmath106 . then @xmath178 , and so @xmath72 is a trefoil knot . since @xmath70 is a meridian of @xmath72 and @xmath70 is isotopic to @xmath101 which in turn is isotopic to @xmath73 , we see that @xmath73 must be a meridian of @xmath72 . if @xmath70 is a @xmath179 curve on @xmath106 then the rolls of @xmath72 and @xmath73 are switched . it is left only to show that @xmath13 and @xmath14 can not both be nonzero . it shall be useful to study the attaching of the exit set of @xmath87 to @xmath106 in terms of the boundary curves of the exit set . they consist of three loops denoted as @xmath180 , @xmath181 and @xmath15 . we take @xmath180 to be isotopic to @xmath70 and @xmath181 to be isotopic to @xmath98 . then @xmath15 is the remaining curve . see figure  [ ktemp.eps ] . our strategy is to show that if @xmath13 and @xmath14 are both nonzero then @xmath15 bounds a disk in @xmath106 and that @xmath15 is a nontrivial knot . this contradiction will then prove our claim . let @xmath182 be the closure of the exit set of @xmath87 . clearly @xmath183 is torus . now @xmath180 and @xmath181 bound an annulus @xmath184 in @xmath185 . thus , @xmath185 has two components , the annulus @xmath184 and another component we shall call @xmath186 which has a single boundary component @xmath15 . now , @xmath187 is a torus with a disk removed . since @xmath188 must be a torus , @xmath186 is a disk . since this torus is embedded in @xmath0 it follows that @xmath15 , the boundary of @xmath186 , is unknotted . the alexander polynomial calculations in lemma 11 show that @xmath72 is knotted , and thus even for the @xmath189 case @xmath180 and @xmath181 are nontrivial knots . now since @xmath180 and @xmath181 are parallel knots in @xmath106 we can deform @xmath87 to appear as in figure  [ ktemp.eps ] . by studying figure  [ gamma ] we see that @xmath15 is a satellite of @xmath180 . this implies @xmath15 is nontrivial and completes the proof of theorem 9 . @xmath152    in any lorenz - smale flow there is a pair of unlinked saddle orbits .    in the lorenz template there are two fixed points in the first return map of the branch line . these correspond to a pair of unlinked unknots in the semi - flow if the lorenz template is standardly embedded . in all of the embeddings allowed for in theorem  [ thm_ls ] these two orbits remain unlinked . thus , by the birman - williams template theorem @xcite , the saddle set also contains a pair of unlinked orbits . @xmath152    consider a smale flow on @xmath0 with an attracting fixed point , a repelling fixed point , a lorenz saddle set and no other basic sets . there is only one possible configuration , and in it the template of the saddle set is a standardly embedded lorenz template .    the proof is similar to case 1 above . wada s classification theorem for morse - smale flows is based on applying a series of _ moves _ to one or two existing morse - smale flows and building up new ones . conclusion ( a ) of the next theorem establishes an operation that produces a new smale flow from two existing ones that is similar to wada s move iv @xcite . [ thm_sums ] let @xmath190 and @xmath191 be nonsingular smale flows on @xmath0 such that ( 1 ) they each have only one attracting closed orbit with knot types @xmath5 and @xmath6 respectively , ( 2 ) there is only one repelling closed orbit which is unknotted and is a meridian of the attractor and , ( 3 ) the repellers bound disks whose interiors meet the chain - recurrent sets in a single point . then ( a ) and ( b ) below hold true . \(a ) there exists a nonsingular smale flow on @xmath0 such that there is only one attracting closed orbit which has knot type @xmath192 , and there is only one repelling closed orbit which is unknotted and is a meridian of the attractor . \(b ) there exists a nonsingular smale flow on @xmath0 such that there is only one attracting closed orbit which has knot type @xmath5 , and there is only one repelling closed orbit which has knot type @xmath6 . furthermore , the attractor and the repeller have linking number one , and can be placed into solid tori whose cores are meridians of the each other . the proofs are simple cut and paste arguments . some details are left to the reader . for ( a ) let @xmath193 , @xmath194 , be tubular neighborhoods of the repellers in @xmath195 , @xmath194 , respectively . let @xmath196 , @xmath194 , be the disks described in hypotheses ( 3 ) . thicken up these disks a bit by taking cross product with a small interval @xmath197 $ ] . we require that each @xmath198 meet the corresponding @xmath199 in an unknotted arc . let @xmath200 . choose the signs so that flows enter the thickened disks on the positive sides . see figure  [ fig_cut ] . delete from the 3-sphere of each flow the interior of the union of @xmath193 and @xmath198 , for the corresponding @xmath194 . we now have flows on two cylinders @xmath201 , @xmath194 . see figure  [ fig_paste ] . the boundary of @xmath201 is the union of @xmath202 , @xmath203 , and the annulus @xmath204 . the flow exits @xmath201 on the interior of @xmath202 , for @xmath194 . glue @xmath205 to @xmath206 by identifying @xmath207 with @xmath208 and @xmath209 with @xmath210 . this creates a solid torus , @xmath211 . the flow induced on @xmath211 is inward on @xmath212 . further , the identifications can be chosen so that the flow on @xmath211 has an attracting orbit with knot type @xmath192 , assuming @xmath211 is standardly embedded . it is now clear how to construct the desired flow on @xmath0 . the proof of conclusion ( b ) is similar and in fact simpler and so is left as an exercise . note that hypotheses ( 3 ) is not required . the class of links which can arise in morse - smale flows ( nonsingular on @xmath0 ) is , according to wada s theorem , quite limited . for smale flows it is easy to construct examples in which every knot and link can be realized simultaneously as a saddle orbits . this is a consequence of the existence of _ universal templates _ , templates in which all links all realized as closed orbits , @xcite ; also see @xcite and @xcite . franks has shown that any link can be realized as an attractor of a smale flow ( * ? ? ? * propsition 6.1 ) . in the proof the link is realized as a braid in an unknotted solid torus whose entrance set is its entire boundary . thus given a smale flow with attractor @xmath1 we can replace @xmath1 with any generalized cable of @xmath1 , though a new saddle set will typically be introduced . in @xcite it is shown that given a smale flow @xmath213 with a saddle set modeled by a template @xmath26 containing the closed orbit @xmath1 , there exists another smale flow @xmath214 with the same basic sets as @xmath213 except that @xmath1 is an attractor and and the template @xmath26 has been replaced ( as a model ) with @xmath215 , a template formed by `` surgering '' @xmath26 along @xmath1 ( a standard template operation ) .    turning our attention to simple smale flows we shall use a similar idea to show that given any knot @xmath1 there exists a simple smale flow with attractor @xmath1 and repeller a meridian of @xmath1 . as a corollary of the construction we can give a `` dynamics '' proof that alexander polynomials multiply under connected sums . the template @xmath216 shown in figure  [ univ.eps](a ) is known to contain all knots as periodic orbits @xcite . thus we can suppose @xmath1 has been realized as an orbit in @xmath216 . we shall work with a variation of @xmath216 shown in figure  [ univ.eps](b ) and denoted @xmath211 . it has five `` lorenz ears '' . notice however that the middle ear does not stretch all the way across ; it is to extend only as far as an outer most arc of @xmath1 . ( technically @xmath211 is not a template , but it is still a branched manifold with a semi - flow ) . now consider the rather odd looking object in figure  [ anyk.eps ] . the dark gray circle represents the tubular neighborhood of a repeller . the the light gray tube has the same knot type as @xmath1 ( though only a portion of it is shown ) ; we have only added an extra loop in an outermost strand of @xmath1 . the dark region is a topological ball which meets the light gray tube at a single disk near the cusp of the fourth lorenz ear . their union is a solid torus @xmath65 . the branched manifold @xmath211 has been cut open along @xmath1 and is now a true template @xmath26 ( compare with the proof of theorem a.3.3 in @xcite ) . the boundary of @xmath26 is in the boundary of @xmath65 . we thicken up @xmath26 to get @xmath217 . now we can regard @xmath217 as a neighborhood of a saddle set . its exit set is attached to @xmath65 as required . from the figure we can see that @xmath218 is a solid unknotted torus . thus , we can use a meridian of @xmath1 as a repeller and build up the desired flow . @xmath152      the claim is that given knots @xmath5 and @xmath6 then @xmath219 . by theorem  [ thm_anyk ] there exist smale flows for @xmath5 and @xmath6 that satisfy the hypotheses of theorem  [ thm_sums ] . we use conclusion ( a ) of theorem  [ thm_sums ] to construct a smale flow with attractor @xmath192 and apply ( * ? ? ? * theorem 4.1 ) noting that there are only two saddle sets and hence only two linking matrices needed in the formula given in ( * ? ? ? * theorem 4.1 ) . @xmath152    we have in these last two sections given a variety of tools for building new smale flows from old ones . many other such results could be stated . but , we are nowhere close to developing a calculus of smale flows along the lines of what wada has done for morse - smale flows . indeed we do nt even know if there are any restrictions on the link type of @xmath110 for simple smale flows .
a smale flow is a structurally stable flow with one dimensional invariant sets . 
 we use information from homology and _ template _ theory to construct , visualize and in some cases , classify , nonsingular smale flows in the 3-sphere . 
 flows , knots , smale flows , templates .    _ 
 ams subject classification ( 1991 ) : _ 
 58f25 , 57m25 .