A geometric characterization of the upper bound for the span of the Jones polynomial

arXiv:0907.5374v2 [math.GT] 4 Nov 2010 A geometric characterization of the upper bound for the span of the Jones polynomial J. Gonz´alez-Meneses and P. M. G. Manch´on May 13, 2013 Abstract Let Dbe a link diagram with ncrossings, sAand sBits extreme states and |sAD|(resp. |sBD|) the number of simple closed curves that appear when smoothing Daccording to sA(resp. sB). We give a general formula for the sum |sAD|+|sBD|for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When Dis dealternator connected, the triangulation is especially simple, yielding |sAD|+|sBD|=n+ 2 −2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [14]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al [3], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number kof dealternator crossings for any dealternator connected diagram. Keywords k-almost alternating diagram, circle number, surgery, dealternator connected diagram, dealternator reduced diagram, Jones polynomial, span. 1 Introduction Every link diagram Dhas two related families of circles, sADand sBD, obtained from Dby applying, respectively, A-smoothing or B-smoothing to each of its crossings, as in Figure 1. We denote by |sAD|(respectively |sBD|), the number of circles in sAD(resp. sBD). There is a well known upper bound [8] for the span of the Kauffman bracket hDi, that is, the difference between the extreme degrees of hDi, for a link diagram Dwith ncrossings: span(hDi)≤2n+ 2(|sAD|+|sBD|)−4. We will refer to the sum |sAD|+|sBD|as the circle number of the diagram D. The aim of this paper is to provide a general formula of the circle number, 1 A-smoothing B-smoothing Figure 1: Aand B-smoothing of a crossing characterizing it as the number of faces of an appropriate triangulation of an appropriate surface. For a connected diagram D, its projection yields a triangulation of S2, where we allow the faces to be polygons with at least one edge, not just triangles. As every vertex in this triangulation has valence 4, its faces can be coloured white and black giving a chessboard colouring, which means that any edge is always boundary of both colours. When Dis alternating, the components of sADare the (straightened) boundaries of the (let say) white faces, and the components of sBDare the boundaries of the black faces, hence |sAD|+|sBD|is the total number of faces in the triangulation. We can count the Euler characteristic of the sphere S2: n−2n+ (|sAD|+|sBD|) = 2, therefore |sAD|+|sBD|=n+ 2 for any connected alternating diagram with n crossings. In particular, when Dis connected and alternating, span(hDi)≤4n. If D is in addition reduced, then we have the equality span(hDi) = 4n(see [8]). What happens for non-alternating diagrams? To understand the answer, we look at the concept of k-almost alternating diagram, a notion introduced by Adams [2]. A diagram Dis said to be k-almost alternating if it has a set of k crossings (called dealternators), and not less than k, such that Dis alternating if we switch all these crossings. Every diagram is k-almost alternating for exactly one k≥0. Of course, the 0-almost alternating diagrams are the alternating diagrams. A 1-almost alternating diagram is just called an almost alternating diagram. Other two definitions are required. To simplify notation, we will identify each crossing of Dwith its corresponding point in the projection, hence every dealternator is identified with a point of S2. The diagram Dis called dealternator connected [3] if there is no simple closed curve in S2intersecting (transversely) the projection of Din a nonempty set of dealternators. Equivalently, each diagram Di(i= 1,...,2k) obtained by smoothing all kdealternators in every possible way, is connected. A diagram Dis called dealternator reduced [3] if there is no simple closed curve in S2intersecting (transversely) the projection of Din exactly one non-dealternator crossing and possibly in some dealternators. Equivalently, each diagram Di(i= 1,...,2k) obtained by smoothing all 2 kdealternators in every possible way, is reduced. In [14, Theorem 4], Zhu proves that span(hDi)≤4(n−k) if Dis a dealternator connected k-almost alternating diagram with ncrossings. In [3, Theorem 4.4], Adams et al. proved that span(hDi)≤4(n−k−2) if Dis a dealternator connected and dealternator reduced k-almost alternating diagram (k≥1) with ncrossings. Historically, this result was proved before Zhu’s theorem above. Indeed, all these results provide the corresponding upper bound for the span of the Jones polynomial VL(t) of the link Lrepresented by D, since span(hDi) = 4 span(VL(t)). The main achievement of this paper is to give a geometrical interpretation of the circle number |sAD|+|sBD|of a k-almost alternating diagram, as the number of faces of an appropriate triangulation of an appropriate surface with boundary and Euler characteristic 2 −3k. This construction generalizes the situation described above for alternating diagrams, and provides nice geometric proofs of the results of Zhu and Adams. We remark that, in [13], Turaev followed a similar topological approach in order to count the circle number in the case of alternating diagrams, using a different surface, sometimes called the Turaev surface in the literature. See [6, Section 9.4] for a nice synthesis of his work. In [7, Corollary 7.3], Dasbach et al. proved that span(hDi)≤4(n−g) where gis the genus of the corresponding Turaev surface. In general, g≤kfor a k-almost alternating diagram [1]. In the case of dealternator connected diagrams we will prove that g=k, in light of which the result of Zhu would also follow from [7, Corollary 7.3]. The paper is organized as follows: In Section 2 we construct a surface S associated to a k-almost alternating diagram, and a suitable graph ΓDin S. In Section 3 we see that if Dis dealternator connected, the graph ΓDdetermines a triangulation of S. If Dhas ncrossings, this immediately yields the equality |sAD|+|sBD|=n+ 2 −2k(Theorem 1), obtaining a simple geometric proof of the result of Zhu. From this we deduce the result of Adams et al. [3], using a simple argument by induction. This is done in Section 4. The general case is treated in Section 5: If S\ΓDhas rconnected components (that we will call regions) and sis the rank of its first homology group, we show that |sAD|+ |sBD|=r+s(Theorem 7). We also give a formula for the circle number in terms of the number of regions (r), crossings (n) and dealternators (k). Namely |sAD|+|sBD|= 2k+ 2r−n−2 (Theorem 8). We finish with an example of these results applied to a pretzel diagram, specifying how to draw the regions of S\ΓDin the plane. Acknowledgements: We are grateful to Hugh R. Morton for several helpful comments on a previous version of this paper. 3 2 The surface and graph associated to a diagram Suppose that Dis a k-almost alternating diagram. In this section we construct a surface of Euler characteristic 2 −3ksuch that |sAD|+|sBD|is the number of faces for a suitable triangulation. The construction of the surface Sis made by performing the following local surgery to S2around each dealternator of D. Take a small closed disc Oaround a dealternator crossing. Let a, b, c, d be the four points in which the boundary of Ocuts transversally the diagram D, say counterclockwise. The boundary of Ois the union of four arcs ab,bc,cd and da. Consider two copies of the band [0,1]×[0,1]. Delete the interior of the disc Oand glue the two bands, identifying {0} × [0,1] and {1} × [0,1] of the first band with ab and dc respectively, and {0} × [0,1] and {1} × [0,1] of the second band with bc and ad respectively (see Figure 2). Figure 2: Local surgery around each dealternator crossing What we are doing locally around each dealternator crossing is to add a hollow handle minus a disc (see Figure 3), hence we obtain a surface Swhich is the connected sum of ktorus minus the interior of kdiscs (Figure 4). In particular the Euler Characteristic of our surface is 2 −3k. Indeed, before deleting the interior of the discs, we have a genus khandlebody, hence its Euler characteristic is 2 −2k. Deleting the kdiscs, we get a final Euler characteristic 2 −3k. Remark 1. In [13], Turaev followed a similar topological approach in order to count the circle number in the case of alternating diagrams, using a different surface, sometimes called the Turaev surface in the literature (see also [6], Section 9.4). Following [7], the Turaev genus of a diagram Dis by definition the genus of its corresponding Turaev surface. Recall that the projection of the diagram Dis a graph on S2, which determines a triangulation of S2admitting a chessboard colouring. In the surface S, we can define a similar graph, that we denote ΓD, in the following way: Start with the sphere S2and the projection of D. Consider the small circle around a dealternator, along which local surgery will be applied. Recall that this circle intersects the projection of Din four points, a,b,cand d, which are interior 4 Figure 3: Local surgery adds a hollow handle minus a disc Figure 4: Type of the resulting surface, with genus kand kboundary components points of their corresponding edges. Now remove the disc and glue the two bands as explained above. We complete the graph ΓDby considering a,b,cand das vertices, and adding four edges corresponding to the segments [0,1] × {i} for i= 0,1 (see Figure 5). In other words, the union of the four new vertices and the four new edges is precisely the boundary component of Scorresponding to the given dealternator. a b c d Figure 5: Each dealternator produces 3 extra vertices and 4 extra edges In particular, it follows that the number of vertices in ΓDis n+ 3k, and the 5 number of edges is 2n+ 4k. Notice that the obtained graph ΓDdoes not yield, in general, a triangulation of S, since the resulting regions are not necessarily homeomorphic to a disc: a property which is equivalent to Dbeing dealternator connected. This observation will allow us to obtain a very simple proof of Zhu’s result [14], as we will see in Section 3. If Dis dealternator connected and in addition dealternator reduced, our construction will also give a simple proof of the result of Adam et al. [3], which will be seen in Section 4. 3 When Dis a dealternator connected diagram If Dis a dealternator connected diagram, the construction of Sand of ΓD immediately determines the circle number in terms of the number of crossings and dealternators. Theorem 1. If Dis a dealternator connected, k-almost alternating diagram with ncrossings, then |sAD|+|sBD|=n+ 2 −2k. Proof. The definition of dealternator connected diagram means precisely that each region determined on the surface Sby ΓDis a disc. In other words, ΓDdetermines a triangulation of S, whose number of faces is precisely the circle number of D. Therefore, since the number of vertices is n+ 3k, the number of edges is 2n+ 4k, and the Euler characteristic of Sis 2 −3k, we immediately obtain the formula: (n+ 3k)−(2n+ 4k) + (|sAD|+|sBD|) = 2 −3k, from which the result follows. This implies the result of Zhu mentioned in the introduction. Corollary 2. [14, Theorem 4] If Dis a dealternator connected, k-almost alternating diagram with ncrossings, then span(hDi)≤4(n−k). Proof. It is well known [10] that if we denote M=n+ 2|sAD| − 2 and m=−n−2|sBD|+2, then the maximal (resp. minimal) degree of the Kauffman bracket of the diagram Dis at most M(resp. at least m). Hence span(hDi)≤ M−m= 2n+2(|sAD|+|sBD|)−4. As, by Theorem 1, |sAD|+|sBD|=n+2−2k under our hypothesis, the result follows. Recall that if Lis a link represented by a diagram D, the span of the Kauffman bracket hDiof Dis four times the span of the Jones polynomial VL(t) 6 of L. Hence, if Lis a link represented by a dealternator connected, k-almost alternating diagram with ncrossings, then span(VL(t)) ≤n−k. We finish this section by proving that the Turaev genus (see Remark 1) agrees with the dealternating number for dealternator connected diagrams. Precisely, Corollary 3. If Dis a dealternator connected k-almost alternating diagram, then its Turaev genus gis equal to k. Proof. It is well known [6] that 2g= 2 + n−(|sAD|+|sBD|) where gis the genus of the Turaev surface built from D. The result follows then from Theorem 1. Remark 2. In light of Corollary 3, the result of Zhu is also a consequence of [7, Corollary 7.3]. 4 When Dis both dealternator connected and dealternator reduced The result of Adams cited in the introduction [3, Theorem 4.4], was originally proved by writing the Kauffman bracket of Din terms of the Kauffman brackets of the connected, reduced and alternating diagrams Di,i= 1 . . . , 2k. Here we will deduce it from the result of Zhu, using a simple argument by induction. In order to start induction, we need a result for adequate diagrams [9]. Theorem 4. [9, Proposition 1] Let Dbe an adequate diagram with ncrossings. Then the terms of the highest and lowest degrees in its Kauffman bracket hDi are (−1)|sAD|−1AMand (−1)|sBD|−1Am, where M=n+ 2|sAD| − 2and m=−n−2|sBD|+ 2. We recall that the number M(resp. m) above is the maximal (resp. minimal) possible degree of the Kauffman bracket hDiof an arbitrary diagram D. Moreover, the degree of any term in the Kauffman bracket is congruent with m(and also with M) modulo 4 (see, for instance, [10]). In other words, the Kauffman bracket of any diagram Dcan be written as hDi=amAm+am+4Am+4 +···+aM−4AM−4+aMAM,(1) where some of the coefficients could possibly be zero. We will call aM(resp. am) the hypothetic maximal (resp. minimal) coefficient of hDi. For any diagram the values of these coefficients are am= (−1)|sBD|−1I(GD B) and aM= (−1)|sAD|−1I(GD A), where GD Band GD Aare certain graphs, and I(G) 7 denotes certain independence number of the graph G(see [10] for details). It turns out that a diagram Dis adequate if and only if both graphs GD Band GD Aare empty, which is a nice characterization of adequacy in terms of graphs –compare to [12, Proposition 2 (ii)]. Since the independence number of the empty graph is one, this gives another proof of Theorem 4. In the particular case we are interested in, the hypothetic extreme coefficients of hDican be described in terms of simpler diagrams, as follows: Lemma 5. Let Dbe a dealternator connected k-almost alternating diagram with ncrossings. Suppose that k > 0and choose in Da dealternator crossing. Let D1(resp. D2) be the diagram obtained by A-smoothing (resp. B-smoothing) this dealternator crossing. Then |sAD1|=|sAD|,|sAD2|=|sAD|+ 1,|sBD1|=|sBD|+ 1 and |sBD2|=|sBD|. Moreover, let aM(resp. aM1,aM2) be the hypothetic maximal coefficient of hDi(resp. hD1i,hD2i). Let am(resp. am1,am2) be the hypothetic minimal coefficient of hDi(resp. hD1i,hD2i). Then aM=aM1+aM2and am=am1+am2. Proof. The equalities |sAD1|=|sAD|and |sBD2|=|sBD|are obvious, so let us prove that |sAD2|=|sAD|+ 1. Of course |sAD2|=|sAD|+ǫ, where ǫ=±1, hence |sAD2|+|sBD2|=|sAD|+|sBD|+ǫ. Since Dis a dealternator connected k-almost alternating diagram with ncrossings, by Theorem 1 we know that |sAD|+|sBD|=n+ 2 −2k. Now notice that D2is a (k−1)-almost alternating diagram with n−1 crossings. Indeed, switching the other k−1 dealternator crossings in D2is equivalent to first switching all the kdealternator crossings of Dand then smoothing the selected dealternator, and any alternating diagram is still alternating after (A or B)-smoothing any crossing. Since D2is also dealternator connected, by Theorem 1 again |sAD2|+|sBD2|= (n−1) + 2 −2(k−1). It follows that (n−1) + 2 −2(k−1) = n+ 2 −2k+ǫ hence ǫ= 1. The equality |sBD1|=|sBD|+ 1 is shown in the analogous way. In order to show that aM=aM1+aM2, recall that hDi=AhD1i+A−1hD2i, so we need to show that M1=M−1 and M2=M+ 1. As D1and D2 are diagrams with n−1 crossings, this is equivalent to |sAD1|=|sAD|and |sAD2|=|sAD|+ 1, so we are done. An analogous argument gives the equality involving am. We can now show in a simpler way the result by Adam et al. 8 Corollary 6. [3, Theorem 4.4] If Dis a dealternator connected and dealternator reduced k-almost alternating diagram with ncrossings, and k > 0, then span(hDi)≤4(n−k−2). Proof. By Theorem 1, we know that the hypothetical maximal value of span(hDi) is M−m= 2n+ 2(|sAD|+|sBD|)−4 = 4(n−k). But as the Kauffman bracket has the expression (1) above, it follows that the above bound will decrease by 8 if we show that aM=am= 0. We proceed by induction on k. Suppose that k= 1, that is, Dhas only one dealternator. Denote D1(resp. D2) the diagram obtained by A-smoothing (resp. B-smoothing) this dealternator crossing. By Lemma 5, one has aM= aM1+aM2. But D1and D2are alternating, reduced diagrams, thus they are adequate [9]. Hence Theorem 4 and Lemma 5 tell us that aM1= (−1)|sAD1|−1= (−1)|sAD|−1, and on the other hand aM2= (−1)|sAD2|−1= (−1)|sAD|. Therefore aM=aM1+aM2= 0. The analogous argument shows that am=am1+am2= 0, so the case k= 1 holds. Suppose now that k > 1 and that the result holds for diagrams with less than kdealternators. Choose one dealternator of Dand apply A-smoothing (resp. B-smoothing) to create the diagram D1(resp. D2). Notice that both D1and D2are dealternator connected and dealternator reduced (k−1)-almost alternating diagrams with n−1 crossings. By induction hypothesis, aM1= am1=aM2=am2= 0. Hence aM=aM1+aM2= 0 and am=am1+am2= 0, so the result follows. 5 The general case If a diagram Dis not dealternator connected, the graph ΓDdoes not determine a triangulation of the surface S, since at least one of the regions determined by ΓDis not homeomorphic to a disc. Nevertheless, these regions (the connected components of S\ΓD) admit a black and white colouring which extends the chessboard colouring of S2: It suffices to colour the bands attached during the surgery in the natural way. Notice that, with this colouring, the components of sADare the boundaries of the white regions, and the components of sBDare the boundaries of the black regions. Notice also that these regions have genus 0 (they are discs with holes), so the number of components of their boundary is determined by the rank of their first homology group (the number of holes). Therefore, the circle number |sAD|+|sBD|is determined by the number of regions in S\ΓD, together with the ranks of their first homology groups. More precisely: Theorem 7. Let Dbe a k-almost alternating diagram with ncrossings, and let Sand ΓDbe defined as in Section 2. Let R1,...,Rrbe the connected components 9