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the fundamental group of the complement of plane curves is a very important topological invariant , which can be also computed for line arrangements . we list here some applications of this invariant . chisini @xcite , kulikov @xcite and kulikov - teicher @xcite have used the fundamental group of complements of branch curves of generic projections in order to distinguish between connected components of the moduli space of smooth projective surfaces , see also @xcite . moreover , the zariski - lefschetz hyperplane section theorem ( see @xcite ) states that @xmath3 where @xmath4 is an hypersurface and @xmath5 is a generic 2-plane . since @xmath6 is a plane curve , the fundamental groups of complements of curves can be used also for computing the fundamental groups of complements of hypersurfaces in @xmath7 .    a different need for fundamental groups computations arises in the search for more examples of zariski pairs @xcite . a pair of plane curves is called _ a zariski pair _ if they have the same combinatorics ( to be exact : there is a degree - preserving bijection between the set of irreducible components of the two curves @xmath8 , and there exist regular neighbourhoods of the curves @xmath9 such that the pairs @xmath10 are homeomorphic and the homeomorphism respects the bijection above @xcite ) , but their complements in @xmath11 are not homeomorphic . for a survey , see @xcite . it is also interesting to explore new finite non - abelian groups which serve as fundamental groups of complements of plane curves in general , see for example @xcite . an arrangement of lines in @xmath12 is a union of copies of @xmath13 in @xmath12 . such an arrangement is called _ real _ if the defining equations of the lines can be written with real coefficients , and _ complex _ otherwise . note that the intersection of the affine part of a real arrangement with the natural copy of @xmath14 in @xmath12 is an arrangement of lines in the real plane .    for real and complex line arrangements @xmath0 , fan @xcite defined a graph @xmath15 which is associated to its multiple points ( i.e. points where more than two lines are intersected ) : given a line arrangement @xmath0 , the graph @xmath15 of multiple points lies on @xmath0 . it consists of the multiple points of @xmath0 , with the segments between the multiple points on lines which have at least two multiple points . note that if the arrangement consists of three multiple points on the same line , then @xmath15 has three vertices on the same line ( see figure [ graph_gl](a ) ) . if two such lines happen to intersect in a simple point ( i.e. a point where exactly two lines are intersected ) , it is ignored ( and the lines are not considered to meet in the graph theoretic sense ) . see another example in figure [ graph_gl](b ) ( note that this definition gives a graph different from the graph defined in @xcite ) . fan @xcite proved some results concerning the projective fundamental group :    [ fan ] let @xmath0 be a complex arrangement of @xmath16 lines and@xmath17 be the set of all multiple points of @xmath0 . suppose that @xmath18 , where @xmath19 is the first betti number of the graph @xmath15 ( hence @xmath18 means that the graph @xmath15 has no cycles ) . then : @xmath20 where @xmath21 is the multiplicity of the intersection point @xmath22 and @xmath23 .    in @xcite , similar results were achieved for the affine and projective fundamental groups by different methods . fan @xcite has conjectured that the inverse implication is also correct , i.e. if the fundamental group @xmath24 can be written as a direct sum of free groups and infinite cyclic groups , then the graph @xmath15 has no cycles .    in an unpublished note , fan @xcite shows that if the fundamental group of the affine complement is a free group , then the arrangement consists of parallel lines . recently , eliyahu , liberman , schaps and teicher @xcite proved fan s conjecture completely . these results motivate the following definition :    let @xmath25 be a fundamental group of the affine or projective complements of some line arrangement with @xmath16 lines . we say that @xmath25 has _ a conjugation - free geometric presentation _ if @xmath25 has a presentation with the following properties :    * in the affine case , the generators @xmath26 are the meridians of lines at some far side of the arrangement , and therefore the number of generators is equal to @xmath16 . * in the projective case , the generators are the meridians of lines at some far side of the arrangement except for one , and therefore the number of generators is equal to @xmath27 . * in both cases , the relations are of the following type : @xmath28 where @xmath29 is an increasing subsequence of indices , where @xmath30 in the affine case and @xmath31 in the projective case . note that for @xmath32 we get the usual commutator . note that in usual geometric presentations of the fundamental group , most of the relations have conjugations ( see section [ mt ] ) . based on the last definition , fan s result yields that if the graph associated to the arrangement is acyclic , then the corresponding fundamental group has a conjugation - free geometric presentation .    the following natural problem arises :    which line arrangements have a fundamental group which has a conjugation - free geometric presentation ? the aim of this paper is to attack this problem . the importance of this family of arrangements is that the fundamental group can be read directly from the arrangement or equivalently from its incidence lattice ( where the _ incidence lattice _ of an arrangement is the partially - ordered set of non - empty intersections of the lines , ordered by inclusion , see @xcite ) without any computation . hence , for this family of arrangements , the incidence lattice determines the fundamental group of the complement . we start with the easy fact that there exist arrangements whose fundamental groups have no conjugation - free geometric presentation : the fundamental group of the affine ceva arrangement ( also known as the _ braid arrangement _ , appears in figure [ ceva ] ) has no conjugation - free geometric presentation . this fact was checked computationally by a package called _ testisom _ @xcite , which looks for isomorphisms ( or proves a non - isomorphism ) between two given finitely - presented group . note that the ceva arrangement is the minimal arrangement ( with respect to the number of lines ) with this property . our main result is :    the fundamental group of following family of real arrangements have a conjugation - free geometric presentation : an arrangement @xmath0 , where @xmath15 is a union of disjoint cycles of any length , has no line with more than two multiple points , and the multiplicities of the multiple points are arbitrary . we also give the exact group structure ( by means of a semi - direct product ) of the fundamental group for an arrangement of @xmath1 lines whose graph is a cycle of length @xmath2 ( i.e. a triangle ) , where all the multiple points are of multiplicity @xmath2 :    let @xmath0 be the real arrangement of @xmath1 lines , whose graph consists of a cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . moreover , it has no line with more than two multiple points . then : @xmath33 where @xmath34 is the free product . as mentioned above , for the family of arrangements with a conjugation - free geometric presentation of the fundamental group , the incidence lattice of the arrangement determines its fundamental group . there are some well - known families of arrangements whose lattice determines the fundamental group of its complement - the families of _ nice _ arrangements ( jiang - yau @xcite ) and _ simple _ arrangements ( wang - yau @xcite ) . it is interesting to study the relation between these families and the family of arrangements whose fundamental groups have conjugation - free geometric presentations , since for the latter family , the lattice determines the fundamental group of the complement too . we have the following remark :    the fundamental group of the arrangement @xmath35 ( appears in figure [ a_n ] ) has a conjugation - free geometric presentation ( this fact was checked computationally ) , but this arrangement is neither nice nor simple .    it will be interesting to find out whether our family of arrangements is broader than the family of simple arrangements , or whether there exists a simple arrangement whose fundamental group has no conjugation - free geometric presentation . [ rem_dehornoy ] it is worth to mention that conjugation - free geometric presentations are complemented positive presentations ( defined by dehornoy @xcite , see also @xcite ) . some initial computations show that in general conjugation - free geometric presentations are not complete ( since the cube condition is not satisfied for some triples of generators ) . nevertheless , we do think that there exist conjugation - free geometric presentations which are complete and hence have all the good properties induced by the completeness ( see the survey @xcite ) . we will discuss this subject in a different paper . the paper is organized as follows . in section [ mt ] , we give a quick survey of the techniques we are using throughout the paper . in section [ length_three ] , we show that the fundamental group of a real arrangement whose graph has a unique cycle of length 3 has a conjugation - free geometric presentation . in this section , we also deal with the exact structure of the fundamental group of a real arrangement whose graph consists of a cycle of length @xmath2 , where all the multiple points have multiplicity @xmath2 . section [ length_n ] deals with the corresponding result for a real arrangement whose graph has a unique cycle of length @xmath16 . we also generalize this result for the case of arrangements whose graphs are a union of disjoint cycles . in this section , we present the computation of the fundamental group of the complement of real line arrangements . this is based on the moishezon - teicher method @xcite and the van kampen theorem @xcite . some more presentations and algorithms can be found in @xcite . if the reader is familiar with this algorithm , he can skip this section .      to an arrangement of @xmath36 lines in @xmath14 one can associate a _ wiring diagram _ @xcite , which holds the combinatorial data of the arrangement and the position of the intersection points . a wiring diagram is a collection of @xmath36 wires ( where a _ wire _ in @xmath37 is a union of segments and rays , homeomorphic to @xmath38 ) . the induced wiring diagram is constructed by choosing a new line ( called the _ guiding line _ ) , which avoids all the intersection points of the arrangement , such that the projections of intersection points do not overlap . then , the @xmath36 wires are generated as follows . start at the @xmath39 end of the line with @xmath36 parallel rays , and for every projection of an intersection point , make the corresponding switch in the rays , as in figure [ latowd ] .    to a wiring diagram , one can associate a list of _ lefschetz pairs_. any pair of this list corresponds to one of the intersection points , and holds the smallest and the largest indices of the wires intersected at this point , numerated locally near the intersection point ( see @xcite and @xcite ) .    for example , in the wiring diagram of figure [ wdtolp ] , the list of lefschetz pairs is ( we pass on the intersection points from right to left ) : @xmath40}},{{\left[{2},{4}\right]}},{{\left[{1},{2}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right]}},{{\left[{3},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right ] } } ) .\ ] ]      let @xmath41 be a closed disk in @xmath14 , @xmath42 a set of @xmath36 points , and @xmath43 . let @xmath44 be the group of all diffeomorphisms @xmath45 such that @xmath46 is the identity and @xmath47 . the action of such @xmath48 on the disk applies to paths in @xmath41 , which induces an automorphism on @xmath49 . the _ braid group _ , @xmath50 $ ] , is the group @xmath44 modulo the subgroup of diffeomorphisms inducing the trivial automorphism on @xmath49 . an element of @xmath51 $ ] is called a _ braid_. for simplicity , we will assume that @xmath52 , and that @xmath53 . choose a point @xmath54 ( for convenience we choose it to be below the real line ) . the group @xmath55 is freely generated by @xmath56 , where @xmath57 is a loop starting and ending at @xmath58 , enveloping the @xmath59th point in @xmath60 . the set @xmath61 is called a _ geometric base _ or _ g - base _ of @xmath55 ( see figure [ fig_1 ] ) . let @xmath62}},\dots,{{\left[{a_p},{b_p}\right]}})$ ] be a list of lefschetz pairs  associated to a real line arrangement @xmath63 with @xmath36 lines . the  of the complement of the arrangement is a quotient group of @xmath55 . there are @xmath64 sets of relations , one for every intersection point . in each point , we will compute an object called a _ skeleton _ , from which the relation is computed .    in order to compute the skeleton @xmath65 associated to the @xmath59th intersection point , we start with an _ initial skeleton _ corresponding to the @xmath59th lefshetz pair @xmath66}}$ ] which is presented in figure [ fig3 ] , in which the points correspond to the lines of the arrangement and we connect by segments adjacent points which correspond to a local numeration of lines passing through the intersection point .    0.7 cm    to the initial skeleton , we apply the lefschetz pairs @xmath67 } } , \cdots,{{\left[{a_1},{b_1}\right]}}$ ] . a lefschetz pair @xmath68}}$ ] acts by rotating the region from @xmath69 to @xmath70 by @xmath71 counterclockwise without affecting any other points . for example , consider the list @xmath72}},{{\left[{2},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{1},{3}\right]}},{{\left[{3},{4}\right]}})$ ] . let us compute the skeleton associated to the 5th point . the initial skeleton for @xmath73}}$ ] is given in figure [ fig4](a ) . by applying @xmath74}}$ ] and then @xmath75}}$ ] , we get the skeleton in figure [ fig4](b ) . then , applying @xmath76}}$ ] yields the skeleton in figure [ fig4](c ) , and finally by acting with @xmath77}}$ ] we get the final skeleton in figure [ fig4](d ) . 4.5 cm      from the final skeletons we compute the relations , as follows . we first explain the case when @xmath78}}$ ] corresponds to a simple point ,  @xmath79 . then the skeleton is a path connecting two points . let @xmath41 be a disk circumscribing the skeleton , and let @xmath60 be the set of points . choose an arbitrary point on the path and pull it down , splitting the path into two parts , which are connected in one end to @xmath80 and in the other to the two end points of the path in @xmath60 . the loops associated to these two paths are elements in the group @xmath81 , and we call them @xmath82 and @xmath83 . the corresponding elements commute in the  of the arrangement s complement . figure [ av_bv ] illustrates this procedure . now we show how to write @xmath82 and @xmath83 as words in the generators @xmath84 of @xmath85 . we start with the generator corresponding to the end point of @xmath82 ( or @xmath83 ) , and conjugate it as we move along @xmath82 ( or @xmath83 ) from its end point on @xmath60 to @xmath58 as follows : for every point @xmath86 which we pass from above , we conjugate by @xmath87 when moving from left to right , and by @xmath88 when moving from right to left . for example , in figure [ av_bv ] , @xmath89 and so the induced relation is : @xmath90    one can check that the relation is independent of the point in which the path is split .    for a multiple intersection point of multiplicity @xmath91 , we compute the elements in the group @xmath81 in a similar way , but the induced relations are of the following type : @xmath92 we choose an arbitrary point on the path and pull it down to @xmath58 . for each of the @xmath91 end points of the skeleton , we generate the loop associated to the path from @xmath58 to that point , and translate this path to a word in @xmath93 by the procedure described above . in the example given in figure [ av_bv_mul ] , we have : @xmath94 , @xmath95 and @xmath96 , so the relations are @xmath97 in this section , we prove the following proposition :    [ triangle - prop ] the fundamental group of a real affine arrangement without parallel lines , whose graph which has a unique cycle of length @xmath2 and has no line with more than two multiple points , has a conjugation - free geometric presentation .    in the first subsection we present the proof of proposition [ triangle - prop ] . the second subsection will be devoted to studying the group structure of the fundamental group of the simplest arrangement of this family .      for simplicity , we will assume that all the multiple points have the same multiplicity @xmath98 , but the same argument will work even if the multiplicities are not equal .    by rotations and translations , one can assume that we have a drawing of an arrangement which has a unique cycle of multiple points of length 3 and has no line with more than two multiple points , as in figure [ arrange_mult_3 ] . we can assume it due to the following reasons : first , one can rotate a line that participates in only one multiple point as long as it does not unite with a different line ( by results 4.8 and 4.13 of @xcite ) . second , moving a line that participates in only one multiple point over a different line ( see figure [ triangle - line ] ) is permitted in the case of a triangle due to a result of fan @xcite that the family of configurations with @xmath1 lines and three triple points is connected by a finite sequence of smooth equisingular deformations . each of the blocks 1,4,5 contains simple intersection points of two pencils .    in block 1 , one can assume that all the intersections of any horizontal line are adjacent , without intervening points from the third pencil . in blocks 4 and 5 , one can assume that all the intersections of any vertical line are adjacent ( in block 4 , the vertical lines are those with positive slopes ) . all the intersection points of block 5 are to the left of all the intersection points of block 4 . hence , we get the list of lefschetz pairs as in table [ tab1 ] ( we put a double line to separate between the pairs related to different blocks ) . @xmath99 &   &   2n(n-1)+3 & [ n , n+1 ] \\ 2 & [ n-1,n ] &   &   2n(n-1)+4 & [ n-1,n ] \\ \vdots & \vdots   &   & \vdots & \vdots \\ n & [ 1,2 ] &   & n(2n-1)+2 & [ 1,2 ] \\ \hline n+1 & [ n+1,n+2 ] & & n(2n-1)+3 & [ n+1,n+2 ] \\ n+2 & [ n , n+1 ] & &   n(2n-1)+3 & [ n , n+1 ] \\ \vdots & \vdots &   & \vdots & \vdots \\ 2n & [ 2,3 ] &   & n(2n)+2 & [ 2,3 ] \\ \hline \vdots & \vdots & & \vdots & \vdots\\ \hline ( n-2)n+1 & [ 2n-2,2n-1 ] &   & ( 3n-1)(n-1)+3 & [ 2n-2,2n-1 ]   \\ ( n-2)n+2 & [ 2n-3,2n-2 ] &   & ( 3n-1)(n-1)+4 & [ 2n-3,2n-2 ] \\ \vdots & \vdots   &   & \vdots & \vdots\\ ( n-1)n & [ n-1,n ] &    & 3n(n-1)+2 & [ n-1,n ] \\ \hline\hline ( n-1)n+1 & [ n,2n ] &   & 3n(n-1)+3 & [ n,2n ] \\ \hline\hline ( n-1)n+2 & [ 2n,3n ] & & & \\ \hline\hline ( n-1)n+3 & [ 2n-1,2n ] &   & & \\ ( n-1)n+4 & [ 2n-2,2n-1 ] & & & \\ \vdots & \vdots   &   & & \\ ( n-1)(n+1)+2 & [ n+1,n+2 ] & & & \\ \hline ( n-1)(n+1)+3 & [ 2n,2n+1 ] & & & \\ ( n-1)(n+1)+4 & [ 2n-1,2n ] & & & \\ \vdots & \vdots &   & & \\ ( n-1)(n+2)+2 & [ n+2,n+3 ] & & &   \\ \hline \vdots & \vdots & & & \\ \hline ( 2n-1)(n-1)+3 & [ 3n-2,3n-1 ] &   & & \\ ( 2n-1)(n-1)+4 & [ 3n-3,3n-2 ] &   & & \\ \vdots & \vdots   &   & & \\ 2n(n-1)+2 & [ 2n,2n+1 ] &   & & \\ \hline \end{array}\ ] ]    by the moishezon - teicher algorithm ( see section [ mt ] ) , we get the following skeletons :    * for point @xmath91 , where @xmath100 , the corresponding final skeleton appears in figure [ braid1 - 6](a ) , where @xmath101 and @xmath102 . * for point @xmath103 , the corresponding final skeleton appears in figure [ braid1 - 6](b ) . * for point @xmath104 , the corresponding final skeleton appears in figure [ braid1 - 6](c ) . * for point @xmath91 , where @xmath105 , the corresponding final skeleton appears in figure [ braid1 - 6](d ) , where @xmath106 and @xmath107 . * for point @xmath91 , where @xmath108 , the corresponding final skeleton appears in figure [ braid1 - 6](e ) , where @xmath109 and @xmath110 . * for point @xmath111 , the corresponding final skeleton appears in figure [ braid1 - 6](f ) . before we proceed to the presentation of the fundamental group , we introduce one notation : instead of writing the relations ( where @xmath22 are words in a group ) : @xmath112 we will sometimes write : @xmath113 $ ] .    by the van kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the line arrangement s complement :    generators : @xmath114 + relations : +    1 .   @xmath115=e$ ] , where @xmath101 and@xmath102 . 2 .   @xmath116 $ ] . 3 .   @xmath117 $ ] . @xmath118=e$ ] where @xmath106 and @xmath107 . 5 .   @xmath119=e$ ] where @xmath109 and @xmath110 . @xmath120 $ ] . now , we show that all the conjugations in relations ( 1),(2),(3 ) and ( 4 ) can be simplified . we start with relations ( 1 ) , and then relations ( 2 ) . we continue to relations ( 4 ) and we finish with relations ( 3 ) . we start with the first set of relations : for @xmath121 , we get that for all @xmath101 we have : @xmath122=e$ ] . now , we proceed to @xmath123 . for @xmath124 , we get : @xmath125=e$ ] . by the relation @xmath126=e$ ] , it is simplified to @xmath127=e$ ] . in this way , we get that for @xmath123 , we have : @xmath128=e$ ] for @xmath101 .    by increasing @xmath129 one by one , we get that all the conjugations disappear and we get @xmath130=e$ ] , where @xmath101 and @xmath102 , as needed .    relations ( 2 ) can be written as : @xmath131 @xmath132 @xmath133 by the simplified version of relations ( 1 ) , we can omit all the generators @xmath134 . hence we get : @xmath135 as needed . we proceed to relations ( 4 ) . we start with @xmath136 . taking @xmath124 , we get : @xmath137=e.\ ] ] by relations ( 6 ) , we have : @xmath138 , and hence we get : @xmath139=e.\ ] ] for @xmath140 , we get : @xmath141=e.\ ] ] by relations ( 6 ) again and the simplified version of relations ( 1 ) , we get : @xmath142=e.\ ] ] using the simplified relation @xmath143=e$ ] , we get @xmath144=e$ ] .    in the same way , we get that for @xmath136 and @xmath145 , we have : @xmath146=e$ ] . we continue to @xmath147 . taking @xmath124 , we have : @xmath148=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath150=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath152=e$ ] .    for @xmath140 , we have : @xmath153=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath154=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath155=e.\ ] ] by @xmath152=e$ ] , we get @xmath156=e$ ] . in the same way , we get that for @xmath147 and @xmath145 , we get : @xmath157=e$ ] .    in the same way , by increasing @xmath129 one by one , we will get that for all @xmath158 and @xmath145 , we get : @xmath130=e$ ] as needed .    relations ( 3 ) can be written : @xmath159 @xmath160 @xmath161 by relations ( 1 ) , we can omit the generators @xmath134 , so we get : @xmath162 @xmath163 @xmath164 by relations ( 2 ) , we can omit also the generators @xmath165 in order to get : @xmath166    hence , we get the following simplified presentation :    generators : @xmath114 + relations :    1 .   @xmath130=e$ ] , where @xmath101 and @xmath102 . 2 .   @xmath167 $ ] . 3 .   @xmath168 $ ] . @xmath130=e$ ] where @xmath106 and @xmath107 . @xmath119=e$ ] where @xmath109 and @xmath110 . 6 .   @xmath120 $ ] . therefore , we have a conjugation - free geometric presentation , and hence we are done . cohen and suciu @xcite give the following presentation of @xmath169 , which is known @xcite to be the fundamental group of the complement of the affine ceva arrangement ( see figure [ ceva ] ) : @xmath170 the actions of the automorphisms @xmath171 and @xmath172 are defined as follows :    @xmath173 + @xmath174    @xmath175 + @xmath176    @xmath177 + @xmath178    notice that if we rotate clockwise the lowest line in the affine ceva arrangement ( figure [ ceva ] ) , we get an arrangement @xmath0 whose graph consists of a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 .    by a simple check , the effect of this rotation is the addition of the commutator relation @xmath179=e$ ] to the presentation of the group . hence , we get that the actions of the automorphisms @xmath171 and @xmath172 are changed as follows :    @xmath173 + @xmath174    @xmath180    @xmath177 + @xmath178    this is the presentation of the group : @xmath181 , where @xmath34 is the free product . to summarize , we get the following result :    let @xmath0 be the arrangement of @xmath1 lines without parallel lines whose graph is a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . then : @xmath33    it is interesting to check how this proposition can be generalize to arrangements whose graphs are cycles of length @xmath182 . in this section , we show that the fundamental group of a real affine arrangement whose graph is a unique cycle of any length and has no line with more than two multiple points , has a conjugation - free geometric presentation . at the end of this section , we generalize this result to arrangements whose graphs are unions of disjoint cycles . we start by investigating the case of a cycle of length @xmath183 and then we generalize it to any length .    in figure [ cycle_multiple5 ] , we present a real arrangement whose graph is a cycle of @xmath183 multiple points ( note that any real arrangement whose graph is a unique cycle of @xmath183 multiple points and has no line with more than two multiple points , can be transferred to this drawing by rotations , translations and equisingular deformations ) .    based on figure [ cycle_multiple5 ] , we get the list of lefschetz pairs presented in table [ tab2 ] .    @xmath184   & 2 & & 13 & [ 6,7 ] & 2 & & 25 & [ 8,9 ] & 2 \\ 2 & [ 5,6 ]   & 2 & & 14 & [ 7,8 ] & 2 & & 26 & [ 6,7 ] & 2 \\ 3 & [ 7,8 ]   & 2 & & 15 & [ 3,4 ] & 2 & & 27 & [ 5,6 ] & 2 \\ 4 & [ 6,7 ]   & 2 & & 16 & [ 4,5 ] & 2 & & 28 & [ 7,8 ] & 2 \\ 5 & [ 4,5 ]   & 2 & & 17 & [ 5,6 ] & 2 & & 29 & [ 6,7 ] & 2 \\ 6 & [ 3,4 ]   & 2 & & 18 & [ 6,7 ] & 2 & & 30 & [ 4,6 ] & 3 \\ 7 & [ 5,6 ]   & 2 & & 19 & [ 4,6 ] & 3 & & 31 & [ 3,4 ] & 2 \\ 8 & [ 4,5 ]   & 2 & & 20 & [ 3,4 ] & 2 & & 32 & [ 4,5 ] & 2 \\ 9 & [ 2,3 ]   & 2 & & 21 & [ 4,5 ] & 2 & & 33 & [ 2,3 ] & 2 \\ 10 & [ 1,2 ] & 2 & & 22 & [ 8,9 ] & 2 & & 34 & [ 1,2 ] & 2 \\ 11 & [ 2,4 ] & 3 & & 23 & [ 7,8 ] & 2 & & 35 & [ 2,4 ] & 3 \\ 12 & [ 4,6 ] & 3 & & 24 & [ 9,10 ] & 2 & &   &   & \\ \hline\hline \end{array}\ ] ]    by the moishezon - teicher algorithm ( see section [ mt ] ) , one can compute the skeletons of the braid monodromy . after the computation , one should notice that actually we can group the intersection points into blocks according to their braid monodromies ( see figure [ cycle_multiple5_block ] ) , since the structure of the skeletons is similar . following this observation , we can deal with each block separately . so , we get the following sets of skeletons :    * quadruples of type q1 : see figure [ quadruple_case1](a ) for @xmath185 . * quadruples of type q2 : see figure [ quadruple_case2](a ) for @xmath186 , @xmath187 , @xmath188 . * a triple of type t1 : see figure [ triple_case1](a ) . * triples of type t2 : see figure [ triple_case2 ] for @xmath189 . * a triple of type t3 : see figure [ triple_case3](a ) . * a triple of type t4 : see figure [ triple_case4](a ) . now we pass to the general case . one can draw an arrangement of @xmath190 lines whose graph is a unique cycle of length @xmath16 and has no line with more than two multiple points in a similar way to the way we have drawn the arrangement of @xmath191 lines whose graph is a cycle of length @xmath183 . hence , one can compute the braid monodromy of the general arrangement in blocks similar to what we have done in the case of @xmath192 :    * quadruples of type q1 : for @xmath193 , see figure [ quadruple_case1](b ) . * quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 , see figure [ quadruple_case2](b ) . * a triple of type t1 : see figure [ triple_case1](b ) . * triples of type t2 : for @xmath196 , see figure [ triple_case2 ] . * a triple of type t3 : see figure [ triple_case3](b ) . * a triple of type t4 : see figure [ triple_case4](b ) .    by the van - kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the complement of the arrangement :    generators : @xmath197 + relations :    * from quadruples of type q1 : 1 . @xmath198=e$ ] where @xmath193 2 . @xmath199=e$ ] where @xmath193 3 . @xmath200=e$ ] where @xmath193 4 . @xmath201=e$ ] where @xmath193 * from quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 : 1 . @xmath202=e$ ] 2 . @xmath203=e$ ] 3 . @xmath204=e$ ] 4 . @xmath205=e$ ] * from the triple of type t1 : 1 . @xmath206=e$ ] 2 . @xmath207=e$ ] 3 . @xmath208 * from triples of type t2 : 1 . @xmath209 where @xmath196 2 . @xmath210=e$ ] where @xmath196 3 . @xmath211=e$ ] where @xmath196 * from the triple of type t3 : 1 . @xmath212 2 . @xmath213=e$ ] 3 . @xmath214=e$ ] * from the triple of type t4 : 1 . @xmath215=e$ ] 2 . @xmath216=e$ ] 3 . @xmath217    we now show that all the conjugations can be simplified , and hence we have a conjugation - free geometric presentation for the fundamental group . we have conjugations in the relations coming from triples of points and quadruples of points . we start with the relations which correspond to triples of points . the conjugation in relation ( 3 ) of the triple of type t1 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t1 . the conjugation in relation ( 1 ) of triples of type t2 can be simplified using relations ( 2 ) and ( 3 ) of the corresponding triples of type t2 . the conjugation in relation ( 3 ) of the triple of type t3 can be simplified using relation ( 3 ) of the triple of type t4 . the conjugation in relation ( 2 ) of the triple of type t3 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t4 . the conjugation in relation ( 1 ) of the triple of type t3 can be simplified using relations ( 2)(3 ) of the triple of type t3 and relations ( 1)(3 ) of the triple of type t4 . we continue to the relations induced by to the quadruples of type q1 . by the first two relations of the quadruples of type q1 , one can easily simplify the conjugations which appear in the last two relations of the quadruples of type q1 . so we get that the relations correspond to the quadruples of type q1 can be written without conjugations . now , we pass to the relations correspond to the quadruples of type q2 . we start with @xmath218 : we have the following relations :    1 . @xmath219=e$ ] 2 . @xmath220=e$ ] 3 . @xmath221=e$ ] 4 . @xmath222=e$ ]    by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , the conjugations in relation ( d ) are canceled and we get @xmath226=e$ ] . by the same relations and the simplified version of relation ( d ) , we get the following relation from relation ( a ) : @xmath227=e$ ] .    substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relation ( c ) becomes @xmath229=e$ ] , and relation ( b ) becomes @xmath230=e$ ] . the same argument holds for any @xmath231 , where @xmath232 and @xmath233 . hence , one can simplify the conjugations in these cases . now , we pass to the case where @xmath234 and @xmath233 . let @xmath235 . we have the following relations :    1 . @xmath236=e$ ] 2 . @xmath237=e$ ] 3 . @xmath238=e$ ] 4 . @xmath239=e$ ]    by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , relations ( a ) and ( d ) become :    1 . @xmath240=e$ ] 2 . @xmath241=e$ ]    by relations ( a ) and ( d ) above , we get @xmath242=e$ ] , and therefore we also get @xmath243=e$ ] . substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relations ( b ) and ( c ) become :    1 . @xmath244=e$ ] 2 . @xmath245=e$ ]    by relations ( b ) and ( c ) above , we get @xmath246=e$ ] and hence @xmath247=e$ ] . it is easy to show by a simple induction that we can simplify the conjugations for any @xmath231 , where @xmath248 and @xmath233 . the remaining case is @xmath249 . we start with @xmath250 . we have the following relations : +    \(a ) @xmath251=e$ ] + ( b ) @xmath252=e$ ] + ( c ) @xmath253=e$ ] + ( d ) @xmath254=e$ ]    we will show that these conjugations can be simplified .    by relation ( 3 ) of triple t4 and relation ( 3 ) of triple t3 , relations ( c ) and ( d ) can be written as : + ( c ) @xmath255=e$ ] + ( d ) @xmath256=e$ ]    by relation ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] , and hence relation ( c ) becomes @xmath259=e$ ] .    by relation ( 1 ) of triple t2 for @xmath257 , we have : @xmath260 and then relation ( d ) becomes : @xmath261=e$ ] . now , we simplify relation ( a ) . again , by relations ( 2 ) and ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] and @xmath262=e$ ] , and hence : @xmath263=e\ ] ] by relation ( 3 ) of triple t4 , we have : @xmath264=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we have : @xmath265=e.\ ] ] by relations ( 1 ) of triple t3 , we finally have : @xmath266=e$ ] . now , we simplify relation ( b ) . by relation ( 3 ) of triple t4 , we have : @xmath267=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we get : @xmath268=e.\ ] ] by relations ( 2 ) and ( 3 ) of triple t3 , we get : @xmath269=e.\ ] ] finally , by relation ( 1 ) of triple t2 for @xmath257 , we have @xmath260 so we get : @xmath270=e$ ] .    by similar tricks , one can simplify the conjugations for all the cases where @xmath249 and @xmath271 . hence , we have a presentation based on the topological generators without conjugations in the relations , and hence we are done .    the above proof is based on the fact that the multiplicity of each multiple point is @xmath2 . we now explain why it can be generalized to any multiplicity . in case of higher multiplicities , the quadruples from the previous case will be transformed to a block of @xmath272 simple points . it can be easily checked that all the conjugations can be simplified in this case . moreover , the triples from the previous case will be transformed into blocks similar to the blocks we had in the case of a cycle of length @xmath2 ( see proposition [ triangle - prop ] ) , and in this case too , it can be easily checked that all the conjugations can be simplified , and hence we have shown that arrangements whose graph is a unique cycle and have no line with more than two multiple points , have a conjugation - free geometric presentation .    using the following decomposition theorem of oka and sakamoto @xcite , we can generalize the result from the case of one cycle to the case of a union of disjoint cycles :    * ( oka - sakamoto ) * let @xmath273 and @xmath274 be algebraic plane curves in @xmath275 . assume that the intersection @xmath276 consists of distinct @xmath277 points , where @xmath278 are the respective degrees of @xmath273 and @xmath274 . then : @xmath279    hence , we have the following result :    if the graph of the arrangement @xmath0 is a union of disjoint cycles of any length and the arrangement has no line with more than two multiple points , then its fundamental group has a conjugation - free geometric presentation . we would like to thank patrick dehornoy , uzi vishne and eran liberman for fruitful discussions . we owe special thanks to an anonymous referee for many useful corrections and advices and for pointing out the connection between our presentations and dehornoy s positive presentations ( remark [ rem_dehornoy ] ) . m. eliyahu , e. liberman , m. schaps and m. teicher , _ characterization of line arrangements for which the fundamental group of the complement is a direct product _ , alg . topo . , to appear . [ available online : http://arxiv.org/abs/0810.5533 ] . d. garber , _ on the connection between affine and projective fundamental groups of line arrangements and curves _ , singularits franco - japonaises ( j .- p . brasselet and t. suwa , eds . ) , sminaires & congrs * 10 * , 6170 ( 2005 ) . d. garber and m. teicher , _ the fundamental group s structure of the complement of some configurations of real line arrangements _ , complex analysis and algebraic geometry , edited by t.  peternell and f .- o . schreyer , de gruyter , 173 - 223 ( 2000 ) . kulikov and m. teicher , _ braid monodromy factorizations and diffeomorphism types _ , izv . nauk ser . mat . * 64*(2 ) , 89120 ( 2000 ) [ russian ] ; english translation : izv . 64*(2 ) , 311341 ( 2000 ) . b. moishezon and m. teicher , _ braid group technique in complex geometry ii : from arrangements of lines and conics to cuspidal curves _ , in algebraic geometry , lect . notes in math . * 1479 * , 131180 ( 1990 ) .
we introduce the notion of a _ conjugation - free geometric presentation _ for a fundamental group of a line arrangement s complement , and we show that the fundamental groups of the following family of arrangements have a conjugation - free geometric presentation : a real arrangement @xmath0 , whose graph of multiple points is a union of disjoint cycles , has no line with more than two multiple points , and where the multiplicities of the multiple points are arbitrary . 
 we also compute the exact group structure ( by means of a semi - direct product of groups ) of the arrangement of @xmath1 lines whose graph consists of a cycle of length @xmath2 , and all the multiple points have multiplicity @xmath2 .