On a conjecture by Kauffman on alternative and pseudoalternating links

On a Conjecture by Kauffman on Alternative and Pseudoalternating Links Marithania Silvero ∗ Departamento de ´ Algebra. Facultad de Matem´aticas. Universidad de Sevilla. Spain. [email protected] March 18, 2015 Abstract It is known that alternative links are pseudoalternating. In 1983 Louis Kauffman conjectured that both classes are identical. In this paper we prove that Kauffman Conjecture holds for those links whose first Betti number is at most 2. However, it is not true in general when this value increases, as we also prove by finding two counterexamples: a link and a knot whose first Betti numbers equal 3 and 4, respectively. Keywords: Alternative links. Homogeneous links. Pseudoalternating links. 1 Introduction In [7] Louis Kauffman defined the family of alternative links, as an extension of the class of alternating links which preserves some of the nice properties of this well-known ∗Partially supported by MTM2010-19355, MTM2013-44233-P, P09-FQM-5112 and FEDER. 1 arXiv:1402.4599v2 [math.GT] 17 Mar 2015 class. Previously, in 1976, E.J. Mayland and K. Murasugi introduced the class of pseudoalternating links [8]. Alternative links are pseudoalternating, and Kauffman conjectured that the converse also holds: Conjecture 1.1. [7] The classes of alternative and pseudoalternating links are identical. Although this conjecture was stated by Kauffman, Mayland and Murasugi posed a similar question in [8]. In this paper we prove Conjecture 1.1 for links having their first Betti number, β, smaller than 3 (this includes the class of knots of genus one), and we provide counterexamples for links whose first Betti numbers equal 3 and 4, respectively (recall that β(L) = min{rank(H1(F))|F is a Seifert surface for L}). Namely, we present a genus two knot and a genus one two-components link which are pseudoalternating but not alternative. The plan of the paper is as follows. In Section 2 we recall the definitions of alternative and pseudoalternating links; we also recall the definition of homogeneous links, an intermediate family introduced by Peter Cromwell in [3]. In Section 3 we disprove Kauffman Conjecture by finding a link and a knot being pseudoalternating and non-homogeneous, hence non-alternative: L9n18{1}and 10145, whose first Betti numbers equal 3 and 4, respectively. Finally, Section 4 is devoted to prove that the Conjecture holds in the case of links having first Betti number smaller than 3, providing an alternative proof of the characterization of homogeneous genus one knots given in [6]; we also give an upper bound for the number of primitive flat surfaces plumbed to construct a generalized one spanning a given pseudoalternating link. Acknowledgements I want to thank Pedro M. G. Manch´on and Juan Gonz´alez-Meneses for telling me about Kauffman Conjecture, which constituted the starting point of this work. I am also grateful for their numerous valuable comments, their suggestions and corrections on preliminary versions of this paper. I would also like to thank J´ozef H. Przytycki for sharing some of his intuitive ideas with me, and to the anonymous referee for his/her many interesting suggestions on an earlier version of this paper. 2 2 Alternative, Homogeneous and Pseudoalternating Links As the alternative, homogeneous and pseudoalternating characters of a link are orientation dependant, from now on all links will be oriented and non-split. Given an oriented diagram Dof a link L, it is possible to smooth every crossing coherently with the orientation of the diagram. After doing this for all crossings in D, we obtain a set of topological (Seifert) circles. Following Kauffman, the spaces of the diagram Dare the connected components of the complement of its Seifert circles in S2, as opposed to the regions of the knot diagram. Draw an edge joining two Seifert circles at the place where there was a crossing in D, and label the edge with the sign of the corresponding crossing (+ or −). We will refer to the resulting set of topological circles and labeled edges as the Seifert diagram of D, because of the analogy of this process to Seifert’s algorithm for constructing an orientable surface spanning a link. Definition 2.1. [7] An oriented diagram Dis alternative if all the edges in any given space of Dhave the same sign. An oriented link is alternative if it admits an alternative diagram. An oriented diagram is alternating if and only if it is alternative and the sign of the edges in its Seifert diagram changes alternatively when passing through adjacent spaces [7, Lemma 9.2]. There are nevertheless alternative links which are not alternating: for instance, every positive (hence alternative) non-alternating link, like the knot 819. We consider now the family of homogeneous links, introduced by Peter Cromwell in 1989 [3]. From the Seifert diagram associated to D, we can construct a graph GDas follows: associate a vertex to each Seifert circle and draw an edge connecting two vertices in GDfor each edge joining the associated circles in the Seifert diagram; each edge must be labeled with the sign + or −of its associated crossing in D. The signed graph GDis called the Seifert graph associated to D. Note that GDcan be obtanined from the Seifert diagram of Dby collapsing each circle to a vertex. Given a connected graph G, a vertex vis a cut vertex if G\{v}is disconnected. A block of Gis a maximal subgraph of Gwith no cut vertices. Blocks of the graph Gcan be thought of in the following way: remove all the cut vertices of G; each remaining connected component together with its adjacent cut vertices is a block of G. 3 Figure 1: Two diagrams Dand D0of the knot 943, their associated Seifert diagrams and Seifert graphs. A dark edge has a positive label; a light edge a negative one. Dshows that 943 is homogeneous, as the diagram in the figure is so; Dis non-alternative, but since D0is so, 943 is alternative. Definition 2.2. [3] A Seifert graph is homogeneous if all the edges of a block have the same sign, for all blocks in the graph. An oriented diagram Dis homogeneous if its associated Seifert graph GDis homogeneous. An oriented link is homogeneous if it admits a homogeneous diagram. Note that the original diagram Dcan be recovered from its Seifert diagram, as the sign and position of the crossings in the diagram are preserved (see Figure 1). However, as the relative position of the circles and the order of the edges is not encoded in the Seifert graph, Dcannot be recovered from GD. Let us finally introduce pseudoalternating links. Starting from an oriented diagram D of a link L, the surface SDobtained by applying Seifert’s algorithm [4] is known as the canonical surface (called projection surface in [4]) of Lassociated to D. The graph GD can also be thought as the spine graph of the corresponding canonical surface. Primitive flat surfaces [8] are those canonical surfaces arising from positive or negative diagrams whose Seifert diagrams have no nested circles. A generalized flat surface is, 4 roughly speaking, an orientable surface obtained by gluing a finite number of primitive flat surfaces along some of their discs. More precisely, given two primitive flat surfaces S1and S2, choose a disc of each one, d1and d2. Now, identify both discs in such a way that there exists a sphere S2⊂S3 separating S3into two non empty 3-balls B1and B2such that Si⊂Biand S2∩Si=di, for i= 1,2. Bands starting at d1and d2are not allowed to overlap when identifying d1 and d2. This special kind of Stallings plumbing (or Murasugi sum) will be noted by ∗ (see Figure 2). Generalized flat surfaces are obtained as a finite iteration of this process, plumbing a primitive flat surface in each step. The first Betti number of a surface S,β(S), is the rank of its first homology group. For a primitive flat surface this is just the number of holes in the surface, or equivalently the number of connected components of the complement of its spine graph in the plane minus 1. As the Euler characteristic of one of these surfaces can be computed as its number of discs dSminus its number of bands bS, it follows that β(S) = bS−dS+ 1. Notice that the first Betti number is additive under this special kind of plumbing: each time one plumbs two surfaces both plumbing discs are identified, so the resulting spine graph can be thought as gluing the two original graphs along a vertex. As generalized flat surfaces are orientable, the following definition makes sense: Definition 2.3. [8] An oriented link is said to be pseudoalternating if it is the boundary of a generalized flat surface, with the natural inherited orientation. The following result implies that the first Betti number of a pseudoalternating link L, β(L), is given by any generalized flat surface spanning it: Proposition 2.4. Among all the connected Seifert surfaces that span a given pseudoalternating link, those being generalized flat surfaces have maximal Euler characteristic, or equivalently, minimal genus. For a proof see [8, Proposition 4.4]. It also follows from a well-known result by Gabai [5, Corollary 6.7]. Given the Seifert diagram of a link, circles containing other circles in both sides (inside and outside) become cut vertices in the associated Seifert graph. As a result, an alternative 5 Figure 2: S1and S2are primitive flat surfaces, with β(S1) = 1 and β(S2) = 4. By an identification of discs d1and d2one obtains the generalized flat surface S=S1∗S2, having β(S) = 5. The link spanned by Sis a pseudoalternating link. diagram is homogeneous, so an alternative link is homogeneous. Let Dbe a homogeneous diagram and SDits projection surface. Let G1, . . . , Gnbe the blocks of the homogeneous Seifert graph GD,Dithe subdiagram of D(together with some arcs) associated to the subgraph Gi⊂GD, and SDi⊂SDthe projection surface constructed from Di. Then SD=SD1∗. . .∗SDn. Since blocks in a graph do not contain cut vertices, each SDiis a primitive flat surface: all bands are twisted in the same way and the discs on the surface are either not nested, or there exists a single disc containing the other ones (and this situation is isotopic to the previous one). Hence, SDis a generalized flat surface. This proves that homogeneous links are pseudoalternating, so Conjecture 1.1 can be restated by saying that the classes of alternative, homogeneous and pseudoalternating links are equal. In the two following sections we will present two pseudoalternating links which are not homogeneous, hence non-alternative; these links are counterexamples to Conjecture 1.1. 6 3 Two counterexamples to Kauffman Conjecture The main problem when trying to deal with Kauffman Conjecture is that identifying whether a link is alternative or pseudoalternating is not easy. Of course, by finding an alternative diagram one shows the alternativity of a link, but this does not help when the link is not alternative. In this sense, working with the family of homogeneous links will help us. Cromwell’s paper [3] is devoted to the study of homogeneous links; he provides some sufficient conditions for determining the non-homogeneous character of a link. In particular, we will use the following result: Theorem 3.1. [3, Corollary 5.1] If Lis a homogeneous link and the leading coefficient of its Conway polynomial ∇(L)is ±1, then the crossing number of Lis at most 2·maxdeg ∇(L). The Conway polynomial of Lis defined as ∇(L) = P(1, z), where P(v, z) is the HOMFLYPT polynomial defined by the skein relation v−1P(L+)−vP(L−) = zP(L0), with normalization P(unknot) = 1. Here L+,L−and L0are links represented by diagrams D+,D−and D0respectively, which only differ in the neighborhood of a point as shown in Figure 3. Figure 3: Diagrams D+,D−and D0. We are ready now to prove the following result: Theorem 3.2. There exists a pseudoalternating link which is not homogeneous. Proof. Consider the oriented link Lwith two components presented by the oriented diagram Dshown in Figure 4. This link is L9n18{1}in [2], which corresponds to the two-components link L9n18 in Thistlethwaite table (or 92 53 in Rolfsen table). Its Conway 7 Figure 4: Dand D0are diagrams of the link L=L9n18{1}in [2]; Sis a Seifert surface for L, and S0is obtained from Sjust by overturning the subsurface S3over the disc d. Finally S0=S1∗S2∗S3. polynomial is ∇(L) = z3+ 4z, so by Theorem 3.1, if Lwere homogeneous its crossing number would be at most 2 ·3 = 6, yielding a contradiction. Consequently, Lis nonhomogeneous. See Figure 4. The diagram Dcan be transformed by a Reidemeister III and a Reidemeister I move into the diagram D0. The new diagram D0allows us to see Las the boundary of a certain surface S. The surface Sis clearly isotopic to S0, and S0is the result of performing two Stallings plumbings of three surfaces, S1, S2, S3, using discs d1,d2and d3as “gluing patches”, this is, S=S1∗S2∗S3. Each of these surfaces consists on two discs joined by a pair of bands twisted in a positive way, so S1, S2and S3are primitive flat surfaces; hence Sis a generalized flat surface and Lis a pseudoalternating link. Since every alternative link is homogeneous, the link L9n18{1}, with β(L) = 3, is a counterexample to Conjecture 1.1. Corollary 3.3. There exists a pseudoalternating link which is not alternative. 8 Note that if one changes the orientation of one component of L(for example, reversing the orientation of the dark component in Figure 4) the resulting link is negative, hence alternative. At this point one can wonder if there exist knots or links which are pseudoalternating and non-alternative with any possible orientation of its components. We will show such an example by finding a knot of genus two with these properties, as pseudoalternating, homogeneous and alternative characters are not orientation-dependant in the case of knots. This knot, whose first Betti number equals 4, is pseudoalternating and non-alternative. It would be interesting to find a link with more than one component being a counterexample to the conjecture with all possible orientations. Theorem 3.4. There exists a pseudoalternating knot which is not homogeneous. Proof. The proof is analogous to that of Theorem 3.2. Let Kbe the genus two knot 10145 in Rolfsen table, with the orientation given in Figure 5. Its Conway polynomial is ∇(K) = z4+ 5z2+ 1, so by Theorem 3.1 we deduce that Kis not homogeneous, as 10 >2·4. See Figure 5. By a finite sequence of Reidemeister moves, the classical diagram representing Kin [1], D1, can be transformed into D4by performing the following steps: from D1, we obtain D2by leaving unchanged the diagram except for the grey strand; perform two Reidemeister III moves on the dotted strand of D2in order to get D3; finally, transform D3into D4by moving the grey strand. At this point it is easy to see that the surface Sbounds K;Sis a generalized flat surface obtained by performing two Stallings plumbing of three primitive flat surfaces (two of them consist on a pair of discs joined by two bands, and the other one consists on two discs together with three bands, as shown in S0) using twice the same disc, d, as “gluing patch”. As a result, Kis a pseudoalternating knot. Corollary 3.5. There exists a pseudoalternating knot which is not alternative. Since 10145 is a non-alternating, non-positive knot of genus two, the following result by Stoimenov provides an alternative proof of Corollary 3.5. We remark that the definition of homogeneous link given by Stoimenov in [9] is equivalent to our definition of alternative link, taken from Kauffman [7]. 9