ON THE DEHN COMPLEX OF VIRTUAL LINKS

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1 ON THE DEHN COMPLEX OF VIRTUAL LINKS RACHEL BYRD, JENS HARLANDER Astrct. A virtul link comes with vriety of link complements. This rticle is concerned with the Dehn spce, pseudo mnifold with oundry, nd the Dehn complex, 2-dimensionl spine of the Dehn spce. In the clssicl cse where the link is plnr, the Dehn spce is the link complement in the 3-sphere. We study topologicl nd geometric properties of the Dehn complex of virtul link. Among other things, we show tht every finitely presented group is the fundmentl group of Dehn complex, nd tht ny lternting triple of n lternting virtul link is non-positively curved squred complex. Keywords: Virtul link, Wirtinger complex, Dehn complex, non-positively squred complex, finitely presented group. Mthemticl Suject Clssifiction 2010: 57M20, 57M05, 20F05 1. Introduction This pper presents results concerning topologicl nd geometric spects of the Dehn complex of virtul links. Virtul knots nd links re n extension of clssicl knot nd link theory nd were first introduced y Kuffmn [6]. A virtul link digrm l is 4-regulr grph in the plne, with over-crossing nd under-crossing informtion t some nodes. The nodes with this dditionl informtion re crossings, nd the remining nodes re clled virtul crossings. The ltter re indicted y circle round the node. Figure 1. A usul crossing nd virtul crossing. From virtul link digrm in the plne, we construct closed, orientle surfce F of miniml genus in which l emeds, so tht only crossings pper s nodes. The virtul crossings dispper ecuse we cn run the edges of the grph over hndles. The detils of this construction re s follows. Strt with the grph nd emed it in R 3 so tht the virtul crossings dispper. We otin 2-mnifold with oundry y thickening every edge of the grph into nd. For every oundry component, glue in disc to otin the desired surfce. We cll the surfce thus otined the projection surfce F of l (see Figure 2). The virtul link digrm in the plne thus leds to link digrm on surfce. 1

2 2 RACHEL BYRD, JENS HARLANDER Figure 2. A virtul link digrm in the plne gives rise to link digrm on surfce, torus, in the cse shown; the originl imge cn e found in Kuffmn [5]. There re vrious link complements ssocited with virtul link. Given virtul link digrm l with projection surfce F. We cn push the digrm into the interior of the thickened surfce F I, nd thus otin n emedding of linked circles in F I. We denote the imge of this emedding y ˆl. If we remove n open neighorhood of ˆl from F I, then we otin compct mnifold with oundry which we denote y M(l). If we cone off the ottom surfce F {0} of M(l), we otin compct pseudo mnifold with oundry, which we cll the Wirtinger spce, W S(l). If we further cone off the top surfce in the Wirtinger spce, we otin nother compct pseudo mnifold with oundry, which we cll the Dehn spce, DS(l). In the clssicl sitution where l is plnr link digrm, M(l) is link complement in the thickened 2-sphere, W S(l) is link complement in the 3-ll, nd DS(l) is link complement in the 3-sphere. The Wirtinger spce nd the Dehn spce hve interesting 2-dimensionl spines, the Wirtinger complex W (l) nd the Dehn complex D(l). The Dehn complex is the min oject of interest for this rticle. We will recll the nture of these spines in the next section. 2. The Dehn Complex Both the Wirtinger nd the Dehn spce, eing pseudo 3-mnifolds with oundry, hve 2-dimensionl spines. The procedure for collpsing the Wirtinger nd the Dehn spce to the Wirtinger complex W (l) nd the Dehn complex D(l) cn e found in Hrlnder/Roserock [4]. We recll the cell structures of these 2-complexes. We egin with the Wirtinger complex. First orient the virtul link digrm l. Lel the under-crossing rcs of l y 1,..., n. The Wirtinger complex hs single vertex. Its edges re in one-to-one correspondence with the under-crossing rcs 1,... n. The 2-cells, lso referred to s fces, re in one-to-one correspondence with the crossings of l. Using the right hnd rule, the orienttion of l

3 ON THE DEHN COMPLEX OF VIRTUAL LINKS 3 yields n orienttion of the oundry of the fces. See Figure 3. The Wirtinger complex provides presenttion of its fundmentl group, the Wirtinger group. We hve π 1 (W (l)) = 1, 2,..., n R 1,..., R m where every reltion, R i, is of the form i j = j k, corresponding to crossing in the virtul knot digrm. i j k Figure 3. The Wirtinger reltion i j = j k otined from crossing in virtul knot digrm. We next give description of the cell structure of the Dehn complex. There re two vertices, v + nd v (which come from the cone points). The edges re in one-to-one correspondence with the connected components, A 1,..., A n, of F l. Ech edge A i is oriented from v + to v. The fces re in one-to-one correspondence with the crossings of l. Considering prticulr crossing x of l. Proceed counterclockwise round the crossing x, strting t the end of n over-crossing, nd red off the four (not necessrily distinct) components encountered, sy A x(1), A x(2), A x(3), nd A x(4). A 2-cell is ttched to the edge-pth, A x(1) A 1 x(2) A x(3) A 1 x(4) (see Figure 4). Without loss of generlity, we my choose the tree of the 1-skeleton of D(l) to e the edge A 1. If we collpse the tree to point, choosing it to e our sepoint d 0, we otin presenttion of the Dehn group, π 1 (D(l)) = A 1,..., A n A1 = 1, A x(1) A 1 x(2) A x(3) A 1 x(4) = 1, for every crossing x l. v + A x(1) A x(4) v A x(1) A x(4) v A x(2) A x(3) A x(2) A x(3) v + Figure 4. A crossing gives rise to 2-cell in the Dehn complex. If l is plnr link digrm, then H 1 (D(l)) = Z m, where m is the numer of circle components of the link. In prticulr the Dehn group of clssicl link is never trivil. The sitution is different for virtul links.

4 4 RACHEL BYRD, JENS HARLANDER 1 A 1 A 1 A 2 A 2 A 2 A 2 A 2 A 2 2 Figure 5. The connected components for the virtul knot digrm re leled A 1 nd A 2 ; under-crossing components re leled 1 nd 2. Exmple 1. Consider the virtul knot digrm k in Figure 5. We hve π 1 (D(k)) = A 1, A 2 A 1 = 1, A 2 A 1 2 A 2A 1 1 = 1, A 2 A 1 2 A 2A 1 1 = 1 = 1. Note lso tht π 1 (W (k)) = 1, = 1 2, 2 1 = 1 1 = Z. A presenttion for the Dehn group cn lso e otined from the Wirtinger complex nd the projection surfce of the virtul link digrm l. Consider the mps F {1} ι W S(l) φ DS(l). The first mp is inclusion, the second mp is topologicl quotient mp induced y coning off the top surfce F {1}. It follows tht φ : π 1 (W (l)) π 1 (D(l)) is surjective nd the kernel is normlly generted y the imge of ι. This gives nother convenient wy to construct the presenttion of π 1 (D(l)) which we descrie next. Orient the virtul link l. Consider n oriented curve c on the projection surfce F. If c intersects n rc of l leled i in positive wy ccording to the right hnd rule, we record n i. If c intersects in negtive wy, we record n 1 i. In this wy the curve c gives rise to word w(c) in the genertors 1,..., n of the Wirtinger presenttion. Theorem 2.1 (Byrd [2]). The fundmentl group of the Dehn complex π 1 (D(l)) is isomorphic to the quotient π 1 (W (l)) / K, where K is normlly generted y words w(c), nd the curves c rnge over set of generting curves of the fundmentl group of the projection surfce F. Exmple 2. We next give n exmple of virtul knot whose Dehn group is non-trivil finite. The Wirtinger group of the virtul knot digrm k in Figure 6 is presented y nd the Dehn group is presented y 1, 2, = 1 3, 3 2 = 2 1, 2 3 = 3 1, A 1, A 2, A 3 A 1 = 1, A 1 A 1 2 A 1A 1 3 = 1, A 1 A 1 2 A 1A 1 3 = 1, A 1 A 1 2 A 1A 1 2 = 1 = Z 2. Hrlnder/Roserock [4] show tht the Wirtinger group of this virtul knot hs torsion. Note tht the projection surfce of k is torus. The curve indicted in Figure 6 gives the

5 ON THE DEHN COMPLEX OF VIRTUAL LINKS 5 reltion 1 = 1 3 in π 1 (D(k)) under the surjective homomorphism ϕ. Adding the reltion from the other generting curve, 1 = 1 2, we cn compute π 1(D(k)) using Theorem 2.1. π 1 (D(k)) = π 1 (W (k)) / { 1 = 1 3, 1 = 1 2 } = Z 2 A 1 A 2 A 3 A 1 1 A 1 A 3 A 2 A A 2 A 1 A 1 A 2 Figure 6. A virtul knot digrm k with Dehn group Z 2. The dshed red line represents generting curve on the projection surfce. 3. Curvture We egin with the definition of reducing circle. Definition 3.1. Let l e virtul link digrm with projection surfce F. A simple, closed curve on F tht intersects l exctly twice is reducing circle. When the reducing circle is seprting, we require tht crossings re contined on oth sides of the seprting reducing circle. Seprting reducing circles led to decompositions of the link digrm. If we cut virtul link digrm l long reducing circle nd connect the four new endpoints with two new smll rcs, we otin two virtul link digrms, l 1 nd l 2, nd l is clled the composition of l 1 nd l 2, l = l 1 #l 2. Cutting long non-seprting reducing circle lso simplifies the virtul link digrm. It results in virtul link digrm whose projection surfce is of smller genus. Definition 3.2. A virtul link digrm is prime if there re no reducing circles, seprting or non-seprting. A squred 2-complex is 2-complex where every 2-cell hs four oundry edges. The Dehn complex nd the Wirtinger complex re exmples of squred 2-complex. Definition 3.3. A squred 2-complex is non-positively curved if there re t lest four squres grouped round ech vertex. Tht is, if every reduced edge cycle in the link grph of ech vertex in the 2-complex hs length t lest four.

6 6 RACHEL BYRD, JENS HARLANDER The following result ws otined y Hrlnder nd Roserock [3]. It extends well known results from clssicl knot theory (see Bridson/Hefliger [1], pge 220). Theorem 3.1. Let l e prime, lternting virtul link digrm. Then the Dehn complex of l is non-postively curved squred complex. We outline the min ides of the proof. One first notes tht since l is lternting, oth vertex links in D(l) re isomorphic to the 1-skeleton of the squre cell decomposition of the projection surfce tht is dul to the one induced y l. In prticulr, ech reduces edge pth in these links hs even length. An edge pth of length two would produce reducing circle. Hence, since l is prime, reduced edge pths in the links hve to hve length t lest four. Thus D(l) is non-positively curved squred complex. Exmple 3. The Dehn complex of the knot digrm on the torus in Figure 7 is nonpositively curved squred complex s there re no reducing circles. Figure 7. A prime lternting knot digrm on torus. Byrd [2] investigted the geometry of the Dehn complex for ritrry lternting virtul links. Theorem 3.2. [2] If l is n lternting virtul link digrm drwn on projection surfce F, then we cn cut long finite numer of reducing circles to otin collection of prime, lternting links, l 1,, l n. This gives decomposition of the Dehn complex D(l) into non-positively curved squred complexes, nd decomposition of the fundmentl group of D(l) into CAT(0) groups. The decomposition of the fundmentl group is not decomposition in the grph of groups sense. It mens tht the fundmentl group of D(l) is quotient of free product of CAT(0) groups nd ech reducing circle introduces reltion of prticulr kind. Exmple 4. The knot digrm in Figure 8, drwn on torus, hs one non-seprting reducing circle shown in red. Hence its Dehn complex is not non-positively curved squred complex. We my pply Theorem 3.2 to eliminte the reducing circle nd otin prime knot digrm on 2-sphere whose Dehn complex is non-positively curved squred complex. The resulting prime knot is shown in Figure 9.

7 ON THE DEHN COMPLEX OF VIRTUAL LINKS 7 Figure 8. An lternting knot digrm on torus with one non-seprting reducing circle, indicted y the red curve. Figure 9. The prime knot otined from the knot shown in Figure 8 y cutting long the red reducing circle. Reducing circles cn lso e eliminted y dding dditionl link components to virtul link digrm. One method is tripling link. We triple virtul link digrm l y dding components which run prllel on either side of ll the originl strnds of the link. Denote the tripled link y l T. An lternting exmple of tripled virtul knot digrm is shown in Figure 10. Figure 10. An lternting triple of n lternting virtul knot digrm.

8 8 RACHEL BYRD, JENS HARLANDER Theorem 3.3. [2] If l T is n lternting triple of ny lternting virtul link digrm l, then D(l T ) is non-positively curved squred complex. 4. Dehn Groups It is well known tht the Dehn complex of clssicl link is sphericl if the components re linked. In prticulr Dehn groups of clssicl links re non-trivil, do not contin torsion, nd do not contin free elin sugroups of rnk higher thn two. We hve lredy given exmples of Dehn groups of virtul links tht cn not occur s Dehn groups of clssicl links. In fct, restrictions of ny kind do not exist for Dehn groups of virtul links. Theorem 4.1. Every finitely presented group is Dehn group. Proof. Let G e group nd X R e finite presenttion of G. If R contins m reltions then consider surfce of genus m. Choose reltion r = x ɛ x ɛ k k from R. Choose hole in the surfce nd plce k circles y 1,...,y k round tht hole, y 1 eing the outermost nd y k eing the innermost circle. Orient the circle y i clockwise if ɛ i = 1 nd counterclockwise if ɛ i = 1. See Figure 11. Figure 11. Oriented circles on surfce of genus 2. If x i = x j, i < j, then we turn the circles y i nd y j into one circle component using hndle. Figure 12 shows the cse ɛ i = 1 nd ɛ j = 1. The other cses re treted similrly. Relle this new component y x i. Notice tht the hndle tht ws ttched introduces new genertors nd reltions in the Dehn group for the circle configurtion (see Theorem 2.1). The two crossings on the hndle coming from the lue under-crossing component (see Figure 12) yield the reltion x i = x i nd its inverse. The meridin of the hndle gives the trivil reltion x i = x i, the longitude gives the reltion = w, where w is the word otined from the circles tht run through the hndle. We cn pply Tietze trnsformtions to remove the genertor nd the reltion = w. The other reltion turns into wx i = x i. We cn use Tietze trnsformtion to remove the genertor nd the reltion wx i = x i. Thus the effect of ttching the hndle on the Dehn group is chnging the reltion y ɛ yɛ i i...yɛ j j...yɛ k k into the reltion y ɛ xɛ i i...xɛ j i...yɛ k k. In the end the reltion y ɛ yɛ k k tht held in the Dehn group of the originl circle configurtion is turned into the reltion

9 ON THE DEHN COMPLEX OF VIRTUAL LINKS 9 y i y j x i Figure 12. Connecting oppositely oriented components using hndle (seen from ove). x i r = x ɛ x ɛ k k. The ttched hndles do not introduce dditionl reltions. We proceed with this construction one reltion t time. Suppose we hve reltion r = x ɛ x ɛ k k nd s = z κ zκ l l nd x i = z j. Then we cn connect the ith circle round the hole for r with the jth circle round the hole for s using the sme hndle construction. In the end we hve produces virtul knot drwn on surfce of genus greter or equl to m with desired Dehn group X R. Figure 13 shows link on torus with Dehn group the Klein ottle group x, y xyx 1 y. Note tht the surfce the virtul link is drwn on is not of miniml genus. One could remedy this y dding n under-crossing component long the meridin of the torus. It is not difficult to check tht dding such n under-crossing component does not chnge the Dehn group. The rgument is the sme s given in the proof of Theorem 4.1, when we dded hndle. x y x y d c Figure 13. A virtul link on torus; the projection surfce hs genus 3, the Dehn group is the Klein ottle group.

10 10 RACHEL BYRD, JENS HARLANDER The generl construction for relizing given presenttion s Dehn presenttion typiclly produces virtul link on surfce of high genus with mny components. Indeed, if we strt with the presenttion x 1,..., x n r 1,..., r m, then the virtul link is drwn on surfce of genus greter or equl to m, nd the link hs greter or equl to n components. Sometimes d hoc constructions yield more efficient virtul links tht relize given group s Dehn group. Theorem 4.2. Every cyclic group Z n is Dehn group of virtul knot with the torus s its projection surfce. Proof of Theorem 4.2. The trivil group (Exmple 1), Z 2 (Exmple 2), nd the infinite cyclic group (the unknot) hve lredy een shown to e Dehn groups of virtul links. For ny n, consider the knot digrm k drwn on torus s in Figure 14. By Theorem 2.1, the Dehn group is π 1 (D(k)) = π 1 (W (k)) / { (n 1)(n 2) n 1 = n 2 (n 1)(n 2) = 1} In π 1 (W (k)), we hve the reltion 2 = 1, or = 1. Continuing long this component (strting t 1 ), we see tht the Wirtinger reltions give = 1 = 2 = = n(n 2). Hence, π 1 (D(k)) = n = 1 = Z n n 2... n n(n 2) (n 1)(n 2)... 2(n 2) (n 2)(n 2) (n 1)(n 2) Figure 14. A knot digrm k on torus with the Dehn group Z n ; the knot runs round the meridin nd longitude n times. Theorem 4.3. Every direct sum of two finite cyclic groups Z m Z n is the Dehn group of virtul link with two components with genus 2 projection surfce. Proof. We cn produce two knots k 1 nd k 2 on tori with Dehn groups Z m = 1 m 1 = 1 nd Z n = 2 n 2 = 1, respectively, using Theorem 4.2. From these two knot digrms, we construct link digrm on genus 2 surfce with Dehn group Z m Z n. Connect the

11 ON THE DEHN COMPLEX OF VIRTUAL LINKS Figure 15. The hndle with the linked components tht connect the two tori. two tori y hndle. Pull strnd of k 1 nd strnd of k 2 onto the hndle nd link the two strnds (see Figure 15). Cll the new link digrm l. If we include the reltions from the new crossing nd the one coming from curves round the new hndle, we otin s the Dehn group of l π 1 (D(l)) = 1, 1, 2, 2 m 1 = n 2 = 1, 2 1 = 1 2, 1 2 = 2 1, 1 = 1, 2 = 2 = 1, 2 m 1 = n 2 = [ 1, 2 ] = 1. Theorem 4.4. Every direct sum of finite cyclic group nd n infinite cyclic group, Z n Z, cn e expressed s Dehn group of virtul link. Proof. Consider knot digrm on torus s in Theorem 4.2 with Dehn group Z n nd the unknot drown on sphere. We cn use the sme construction s in Theorem 4.2. Note tht the connected sum of torus with sphere is torus, we end up with the desired 2-component link on torus. A 3 A 4 A 2 A 1 Figure 16. A 2-component link on torus with the Dehn group Z 3 Z. Figure 16 shows 2-component link l on torus with Dehn group Z 3 Z. We cn lso compute the Dehn group of l directly from the Dehn complex. We otin

12 12 RACHEL BYRD, JENS HARLANDER π 1 (D(l)) = A 1, A 2, A 3, A 4 A 1 = 1, A n 2 = 1, A 1 A 1 3 A 4A 1 2 = 1, A 1 A 1 2 A 4A 1 3 = 1 = Z 3 Z Two simply linked circles provides n oviously exmple of knot digrm with Dehn group Z Z. We hve now shown tht every cyclic group is Dehn group of virtul knot nd every elin group tht is generted y two elements is the Dehn group of virtul 2-component link. References [1] M. R. Bridson nd A. Hefliger. Metric Spces of Non-Positive Curvture. Springer, Berlin, [2] R. E. Byrd. On the geometry of virtul knots. Mster s thesis, Boise Stte University, [3] J. Hrlnder nd S. Roserock. Generlized knot complements nd some sphericl rion disc complements. Journl of Knot Theory nd Its Rmifictions, 12(7): , [4] J. Hrlnder nd S. Roserock. On distinguishing virtul knot groups from knot groups. J. of Knot Theory nd Its Rmifictions, 19(5): , [5] L. H. Kuffmn. Introduction to virtul knot theory. rxiv: v2 [mth.gt], Sumitted. [6] L.H. Kuffmn. Virtul knot theory. Europ. J. Comintorics, 20: , Rchel Byrd The Ohio Stte University yrd.197@osu.edu Jens Hrlnder Boise Stte University jenshrlnder@oisestte.edu