A volumish theorem for alternating virtual links
AA VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS
ABHIJIT CHAMPANERKAR AND ILYA KOFMAN
Abstract.
Dasbach and Lin proved a “volumish theorem” for alternating links. Weprove the analogue for alternating link diagrams on surfaces, which provides bounds onthe hyperbolic volume of a link in a thickened surface in terms of coefficients of its reducedJones-Krushkal polynomial. Along the way, we show that certain coefficients of the 4–variable Krushkal polynomial express the cycle rank of the reduced Tait graph on thesurface. Introduction
In [7], Dasbach and Lin proved the following “volumish” theorem for any hyperbolicalternating knot K in S : Let V K ( t ) = a n t n + · · · + a m t m be the Jones polynomial of K , with sub-extremal coefficients a n +1 and a m − . Let v tet ≈ . v oct ≈ . v oct (max( | a n +1 | , | a m − | ) − ≤ vol( S − K ) ≤ v tet ( | a n +1 | + | a m − | − . Their proof relied on volume bounds proved in [13, 1], which showed that the hyperbolicvolume of S − K is linearly bounded above and below by the twist number t ( K ). Dasbachand Lin proved that for any reduced alternating diagram of K , the twist number t ( K ) = | a n +1 | + | a m − | .Recently in [8, 10], similar linear volume bounds in terms of twist number were provedfor certain alternating links in thickened surfaces, but the twist number was not provedto be a link invariant. For alternating links in S , the invariance of t ( K ) follows from theproof of the Tait flyping conjecture in [14], but the Tait flyping conjecture remains open foralternating virtual links (see [3]).In Section 3 below, for a link K in a thickened surface F × I , we define a homological twistnumber τ F ( K ). In Section 4, we give a sufficient condition for τ F ( K ) to be an invariant of areduced alternating surface link diagram by expressing τ F ( K ) in terms of specific coefficientsof the reduced Jones-Krushkal polynomial. Using the new volume bounds in terms of twistnumber, we prove a “volumish” theorem for alternating links on surfaces, which extends tovirtual links.There is an underlying similarity between the proofs of the two volumish theorems. Foralternating links in S , to prove that the twist number is expressed by the sub-extremalcoefficients of the Jones polynomial, Dasbach and Lin relied on two key facts: (1) the Jonespolynomial of an alternating link is a specialization of the two-variable Tutte polynomialof its Tait graph, and (2) certain coefficients of the Tutte polynomial express the cyclerank of the reduced Tait graph. For alternating links in thickened surfaces, we rely on twosimilar facts: (1) the reduced Jones-Krushkal polynomial is a specialization of the Krushkalpolynomial, which extends the Tutte polynomial to a 4–variable polynomial invariant of a r X i v : . [ m a t h . G T ] N ov A. CHAMPANERKAR AND I. KOFMAN graphs on surfaces, and (2) certain coefficients of the Krushkal polynomial express the cyclerank of the reduced Tait graph on the surface (see Definition 3.1). The latter claim forthe Krushkal polynomial is Theorem 2.3, which is of independent interest, and is proved inSection 2 below.Let J K ( t, z ) denote the reduced Jones-Krushkal polynomial, defined in Section 4 below.Boden and Karimi [3] proved that J K ( t, z ) is an invariant of oriented links under isotopyand diffeomorphism of the thickened surface. In Theorem 4.3, we express the homologicaltwist number in terms of specific coefficients of J K ( t, z ). This provides linear bounds onthe hyperbolic volume of the link K in the thickened surface in terms of the sub-extremalterms of J K ( t,
0) using the following geometric results.For a link K in a thickened surface F × I with a weakly generalized alternating (WGA)diagram, Howie and Purcell [8] defined the twist number t F ( K ) on the projection surface F × { } , and showed there is a lower bound on volume in terms of the twist number. Notethat if F is a torus, then F × I − K has a unique hyperbolic structure; for g ≥
2, weconsider the unique hyperbolic structure for which the boundary surfaces F × {± } aretotally geodesic. A surface link diagram D is cellularly embedded if the regions F − D aredisks. Kalfagianni and Purcell [10] proved there is also an upper bound on volume when K has a cellularly embedded WGA diagram D . In particular, D has representativity at least4 on F . (See [10, Section 2] for definitions.)A crossing c is called nugatory if there exists a separating simple closed curve on F thatintersects D only at c . A surface link diagram D is called reduced if it is cellularly embeddedand has no nugatory crossings. Additionally, D is strongly reduced if there do not exist anysimple closed curves on F that intersect D at only one crossing; i.e., neither Tait graph of D on F has loops. A WGA diagram is reduced alternating, but it may not be stronglyreduced.We now combine the hyperbolicity and lower bound from [8], the upper bound from[10] modified for the homological twist number, and our Theorem 4.3 below to state thevolumish theorem for alternating virtual links: Theorem 1.1.
For a closed orientable surface F of genus g ≥ , let K be a non-splitoriented link in F × I that admits a cellularly embedded, strongly reduced WGA diagram D on F × { } . Let τ F ( K ) be the homological twist number of D . Let J K ( t,
0) = a n t n + · · · + a m t m ,with sub-extremal coefficients a n +1 and a m − . Then τ F ( K ) = | a n +1 | + | a m − | − g,τ F ( K ) is an invariant of K in F × I , and F × I − K is hyperbolic with v oct τ F ( K ) ≤ vol( F × I − K ) < v tet τ F ( K ) if g = 1 ,v oct τ F ( K ) − χ ( F )) ≤ vol( F × I − K ) < v oct τ F ( K ) if g ≥ . We prove Theorem 1.1 in Section 4 below. The strongly reduced condition on D canbe weakened to allow certain loops in the Tait graph if we use the expression for τ F ( D ) inTheorem 4.3. See Corollary 4.4 for cases with loops such that τ F ( D ) is a link invariant. Virtual links.
Virtual links and links in thickened surfaces are compared in detail in [3]. Inshort, virtual links are in one-to-one correspondence with stable equivalence classes of linksin thickened surfaces, and each such class has a unique irreducible representative [12]. Forany virtual link diagram, there is an explicit construction to associate a cellularly embeddedlink diagram on a minimal genus surface. Moreover, a virtual link is alternating if and only
VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 3 if it can be represented by an alternating surface link diagram. Any reduced alternatingsurface link diagram is checkerboard colorable, but alternating virtual links also admitalternating surface diagrams which are not checkerboard colorable. The main result of [3]is the following diagrammatic characterization of alternating links in thickened surfaces: If K is a non-split alternating link in F × I , then any connected reduced alternating diagram D on F has minimal crossing number c ( K ), and any two reduced alternating diagrams of K have the same writhe w ( K ).The main result of [3] then implies that the reduced alternating surface link diagram hascrossing number and writhe that are invariants of the virtual link. By [4, Corollary 8], F isthe minimal genus representative of K . So we obtain an invariant of alternating virtual linksby computing J K ( t, z ) on a minimal genus representative reduced alternating surface linkdiagram D . The genus of F is encoded as the highest power of z in J K ( t, z ). Corollary 4.4below then implies that the homological twist number of D on F is also an invariant ofthe virtual link. Thus, Theorem 1.1 extends to any alternating virtual link that admits anappropriate alternating surface link diagram. Related results.
Recently, several preprints have appeared with related results.In [5], Boden, Karimi and Sikora prove the analogues of the Tait conjectures for adequatelinks in thickened surfaces. Any alternating link diagram in a thickened surface is adequate,so a natural question is how to extend Theorem 4.3 to adequate links in thickened surfaces.In [2], a general equivalence is established between ribbon graphs and virtual links. Asour main results rely on the Krushkal polynomial, which is an invariant of ribbon graphs,this philosophy underlies our results as well.In [9], Bavier and Kalfagianni prove results similar to Theorem 1.1 without using poly-nomial invariants of ribbon graphs. Note that in [9], reduced is the same as strongly re-duced here. Their proof relies on the guts of a 3–manifold cut along an essential surface,which is the union of all hyperbolic pieces in its JSJ-decomposition, and the Euler char-acteristic of the guts is related to the twist number using results in [5]. Significantly, toprove that the twist number is invariant, Bavier and Kalfagianni used another part of theKauffman bracket skein module S ( F × I ), which has a basis of all multi-loops on F , in-cluding ∅ . Let J ( K ) = b n t n + · · · + b m t m be the normalized invariant of K in F × I coming from the coefficient in Z [ A ± ] of ∅ , so just the contractible states on F . Theyproved t F ( K ) = | b n +1 | + | b m − | − g. In contrast, the Jones-Krushkal polynomial J K ( t, F that are null-homologous, including non-contractible states on F . Thus, J K ( t, (cid:54) = J ( K ) if g ≥
2, and in Proposition 3.3 below, we show that τ F ( K ) (cid:54) = t F ( K ) if g ≥
2. For links in thickened surfaces, we prove invariance of the homological twist numberin Corollary 4.4 for more general alternating link diagrams than just strongly reduced onesbecause loops in Tait graphs are allowed, as long as there are no genus-generating loops.
Acknowledgements.
The research of both authors is partially supported by grants fromthe Simons Foundation and PSC-CUNY.2.
The Krushkal polynomial
Krushkal [11] introduced a 4–variable polynomial invariant of a graph G embedded in aclosed orientable surface F . We denote this polyomial by p G ( x, y, u, v ) and refer to it as the Krushkal polynomial . The variables x and y play the same role as in the Tutte polynomial,while u and v reflect how G is embedded on F . If G is cellularly embedded (i.e., the facesof G on F are disks), and G ∗ denotes the dual graph on F , then the Krushkal polynomial A. CHAMPANERKAR AND I. KOFMAN generalizes the Tutte polynomial, satisfying both of its key properties: contraction-deletionand a duality relation, p G ( x, y, u, v ) = p G ∗ ( y, x, v, u ).The Krushkal polynomial is defined as the following sum over spanning subgraphs, suchthat every subgraph contributes a monomial weight x a y b u c v d , where the exponents aretopological quantities related to the embedding of this subgraph. Definition 2.1 ([11]) . Let G be a graph cellularly embedded in a closed orientable surface F . The genus of a subsurface S ⊂ F is the genus of the closed surface obtained from S bycapping off all the boundary components of S by disks. For a spanning subgraph H of G ,let H denote the regular neighborhood of H on F . Let i : G → F denote the embedding,and let i : H → F denote its restriction to H . Define: c ( H ) = number of components of H,s ( H ) = twice the genus of H ,s ⊥ ( H ) = twice the genus of the subsurface F − H ,k ( H ) = dim(ker( i ∗ : H ( H ; R ) → H ( F ; R ))) . The Krushkal polynomial is defined as the following sum over all spanning subgraphs H ⊂ G :(1) p G ( x, y, u, v ) = (cid:88) H ⊂ G x c ( H ) − c ( G ) y k ( H ) u s ( H ) / v s ⊥ ( H ) / . We will refer to the monomial terms in (1) as weights on corresponding subgraphs of G .The Tutte polynomial T G ( X, Y ) is related to the Whitney rank generating function R G ( x, y ) by T G ( X, Y ) = R G ( X − , Y −
1) (see [15, § g denotes the genus of F , by [11,Lemma 2.3],(2) R G ( x, y ) = y g p G ( x, y, y, y − ) , and T G ( X, Y ) = R G ( X − , Y − x = X − y = Y − P G ( X, Y, U, V ) = p G ( X − , Y − , U, V ) . Another specialization to obtain the Jones-Krushkal polynomial is discussed in Section 4.
Definition 2.2.
Two edges in G are parallel if they are homologous on F . Note that parallelnon-loop edges connect the same vertices, but parallel loops may be disjoint. Let G (cid:48) denotethe reduced graph of G obtained by deleting all but one edge in each set of parallel edgesin G , and deleting all homologically trivial loops, such that the vertex set V ( G (cid:48) ) = V ( G ).Let G = ( V, E ) and G (cid:48) = ( V, E (cid:48) ). Let (cid:96) = (cid:96) ( G (cid:48) ) denote the subgraph of loops in G (cid:48) , and let G (cid:48) − (cid:96) = ( V, E (cid:48) − (cid:96) ). Let µ = b ( G (cid:48) − (cid:96) ) = | E (cid:48) − (cid:96) | − | V | + c ( G (cid:48) ) and λ = b ( (cid:96) ) = | (cid:96) | . Note that although G (cid:48) is not uniquely determined, µ and λ are invariants of G . Theorem 2.3.
Let G be a graph embedded in a surface F of genus g ≥ . Let (cid:96) be the set ofhomologically trivial loops in G . Let k = | (cid:96) | and n = | V ( G ) | − c ( G ) . Then P G ( X, Y, U, V ) has the following coefficients: µ V g X n − Y k + λ V g − X n Y k . VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 5
Proof.
By [11, Lemma 2.2], p G ( x, y, u, v ) has the property that if e is a loop in G which istrivial in H ( F ), then p G = (1 + y ) p G − e , so that P G = Y P G − e . Thus, we only need to provethe case | (cid:96) | = 0, so we will consider only loops in G that are non-trivial in H ( F ).The unique spanning subgraph H of G which consists of only vertices and no edges hasweight v g x n . Since any other subgraph has a non-empty edge set, its weight has a lowerexponent of x (if it has non-loop edges), or a lower exponent of v (if it has homologicallynon-trivial loops). Thus, the term v g x n occurs in p G ( x, y, u, v ) with coefficient 1.Let e (cid:48) be a non-loop edge of G (cid:48) , and let { e , . . . , e m } be the set of all edges of G parallelto e (cid:48) , which we call the edge class of e (cid:48) . For 1 ≤ j ≤ m , let H j denote one of the spanningsubgraphs of G which consists of j edges from the edge class of e (cid:48) , and no other edges. Theweight of each H j is v g x n − y j − . Summing over the weights of all such spanning subgraphs { H j ⊂ G } , we get the following contribution to p G ( x, y, u, v ):(3) m (cid:88) j =1 (cid:18) mj (cid:19) v g x n − y j − = v g x n − y m (cid:88) j =1 (cid:18) mj (cid:19) y j = v g x n − y ((1 + y ) m − . Thus, for every non-loop edge e (cid:48) in G (cid:48) , its edge class in G contributes the expression (3) to p G ( x, y, u, v ).If H is a spanning subgraph of G with the factor x n − in its weight, then c ( H ) = | V | − H has the form of some H j , possibly with loops added. If H has any loops, then sincethe loops are homologically non-trivial by assumption, the weight of H has an exponentof v which is strictly less than g . Thus, any term in p G ( x, y, u, v ) with a v g x n − factor iscontributed only by the subgraphs H j , so the term must be v g x n − y j − for j ≥ P G ( X, Y, U, V ). With the substitution x = X − y = Y −
1, the expression (3) simplifies to V g ( X − n − Y − Y m −
1) = V g X n − (1 + Y + . . . + Y m − ) + O ( X n − ) . Every non-loop edge in G (cid:48) contributes such an expression to P G ( X, Y, U, V ). Moreover, asdiscussed above, the weight for H is v g x n , which becomes V g ( X − n . Since v g x n alwayshas coefficient 1 in p G , H contributes an additional coefficient − n to the term V g X n − in P G . Therefore, if | (cid:96) | = 0, the coefficient on V g X n − in P G ( X, Y, U, V ) is | E (cid:48) − (cid:96) | − n = | E (cid:48) − (cid:96) | − | V | + 1 = b ( G (cid:48) − (cid:96) ) = µ. This proves the claim for µ .We now proceed similarly for loops in G (cid:48) . Let f (cid:48) be a loop of G (cid:48) , and let { f , . . . , f m } bethe set of all loops of G parallel to f (cid:48) , which we call the edge class of f (cid:48) . For 1 ≤ j ≤ m ,let L j denote one of the spanning subgraphs of G which consists of j loops from the edgeclass of f (cid:48) , and no other edges. Since we assumed that all loops in G are homologicallynon-trivial, the weight of L j is v g − x n y j − . Summing over the weights of all such spanningsubgraphs { L j ⊂ G } , we get the following contribution to p G ( x, y, u, v ):(4) m (cid:88) j =1 (cid:18) mj (cid:19) v g − x n y j − = v g − x n y m (cid:88) j =1 (cid:18) mj (cid:19) y j = v g − x n y ((1 + y ) m − . Thus, for every loop f (cid:48) in G (cid:48) , its edge class in G contributes the expression (4) to p G ( x, y, u, v ).If H is a spanning subgraph of G with the factor x n in its weight, then c ( H ) = | V | .Hence, H consists of only homologically non-trivial loops. We have three cases: A. CHAMPANERKAR AND I. KOFMAN (a) All loops in H are in one edge class of G (cid:48) ,(b) H has loops in distinct edge classes of G (cid:48) , and g ( H ) = 0,(c) H has loops in distinct edge classes of G (cid:48) , and g ( H ) > H is one of the subgraphs L j . In case (b), H has at least one pair of homologicallynon-trivial and non-homologous loops, so g ( H ) = 0 implies that F − H has genus strictlyless than g −
1. Hence, the weight of H has an exponent of v which is strictly less than g − H has a factor u i with i >
0. Therefore, any term in p G ( x, y, u, v )with a v g − x n factor and without a u factor is contributed only by the subgraphs L j , sothe term must be v g − x n y j − for j ≥ x = X − y = Y −
1, the expression (4) simplifies to(5) V g − ( X − n Y − Y m −
1) = V g − X n (1 + Y + . . . + Y m − ) + O ( X n − ) . Every loop in G (cid:48) contributes such an expression to P G ( X, Y, U, V ), so if | (cid:96) | = 0, thecoefficient on V g − X n in P G ( X, Y, U, V ) is λ . This completes the proof of the theorem. (cid:3) Below, we will need another coefficient of P G ( X, Y, U, V ), using the following definition.
Definition 2.4.
For a graph G on the surface F , let (cid:96) ( G ) be the subgraph of loops in G .We will say that { e , e } ⊂ (cid:96) ( G ) are genus-generating loops if g ( H ( e ∪ e )) >
0. Let G (cid:48) bethe reduced graph of G . Define γ ( G ) = { { e , e } ⊂ (cid:96) ( G (cid:48) ) | g ( H ( e ∪ e )) > } . We will say that { e , e , e } ⊂ (cid:96) ( G ) are 3 –petal loops if no pair of loops is parallel and g ( H ( e ∪ e ∪ e )) > k ( e ∪ e ∪ e ) > . Note that if γ ( G ) = 0, then G has no 3–petal loops. The following figure shows an exampleof a graph with 3–petal loops on the torus: Lemma 2.5.
Let G be a graph embedded in a surface F of genus g , such that G has no –petal loops. Let k = | (cid:96) | , n = | V ( G ) | − c ( G ) , and γ = γ ( G ) . Then P G ( X, Y, U, V ) hasthe following coefficient: γ U V g − X n Y k . Proof.
As in the proof above, it suffices to prove the case k = 0, so we can assume that allloops in G are homologically non-trivial. We now determine all possible H ⊂ G that cancontribute to the term U V g − X n in P G ( X, Y, U, V ). Due to the substitution x = X − y = Y −
1, we need to consider H ⊂ G with weight uv g − x i y j . Since i ≤ n , the factor X n implies that H can contribute to the term U V g − X n only if i = n . Hence, c ( H ) = | V ( G ) | so that H ⊂ (cid:96) ( G ) with weight uv g − x n y j .Let H (cid:48) ⊂ G (cid:48) be the reduced graph of H , as in Definition 2.2. Let H (cid:48) be the regularneighborhood of H (cid:48) in F . The condition that G has no 3–petal loops implies that G (cid:48) andhence H (cid:48) have no 3–petal loops. By [11, Equation (4.7)], k ( H (cid:48) ) + g ( F ) + g ( H (cid:48) ) − g ( F − H (cid:48) ) = b ( H (cid:48) ) . VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 7
Figure 1.
An alternating link diagram (left) and its Tait graph G A (right)on the torus [11, Figure 5].The factor U V g − implies that g ( H (cid:48) ) = 1 and g ( F − H (cid:48) ) = g ( F ) −
1. Thus, k ( H (cid:48) ) = b ( H (cid:48) ) − | E ( H (cid:48) ) | −
2. Since g ( H (cid:48) ) = 1, the condition that H (cid:48) has no 3–petal loops nowimplies k ( H (cid:48) ) = 0, so that | E ( H (cid:48) ) | = 2. So the only possible H (cid:48) ⊂ G (cid:48) are the subgraphs { e ∪ e } ⊂ (cid:96) ( G (cid:48) ) such that g ( H (cid:48) ) = 1. Therefore, if H ⊂ G contributes to the term U V g − X n in P G ( X, Y, U, V ), then H (cid:48) is a pair of genus-generating loops.Let { e (cid:48) ∪ e (cid:48) } ⊂ (cid:96) ( G (cid:48) ) be a pair of genus-generating loops, and suppose for I = 1 , G has m I parallel loops in the edge class e (cid:48) I . Let H i,j ⊂ G denote the subgraph with i loops (resp. j loops) in the edge class e (cid:48) (resp. e (cid:48) ), which has weight uv g − x n y ( i − j − . As in (4),summing over the weights of all H i,j ⊂ G , we get the following contribution to p G ( x, y, u, v ):(6) (cid:88) ≤ i ≤ m ≤ j ≤ m (cid:18) m i (cid:19)(cid:18) m j (cid:19) uv g − x n y ( i − j − = uv g − x n y ((1 + y ) m − y ) m − . Thus, for every pair of genus-generating loops in G (cid:48) , its edge class in G contributes theexpression (6) to p G ( x, y, u, v ). As in (5), with the substitution x = X − y = Y − U V g − X n (1 + Y + . . . + Y m − )(1 + Y + . . . + Y m − ) + O ( X n − ) . Every pair of genus-generating loops in G (cid:48) contributes such an expression to P G ( X, Y, U, V ),so if k = | (cid:96) | = 0, the coefficient on U V g − X n in P G ( X, Y, U, V ) is γ ( G ). (cid:3) The homological twist number
In this section, we introduce the homological twist number τ F ( D ), which counts sets ofhomologically twist-equivalent crossings. In contrast, the usual twist number t F ( D ), definedin [10, Definition 2.4], counts twist regions (maximal strings of bigons) of D on F . Everytwist region contributes one homological twist to τ F ( D ), but some crossings of D which arein distinct twist regions can be homologically twist-equivalent. An important advantageof Definition 3.2 below is that τ F ( D ) is invariant for any reduced alternating surface linkdiagram D , without the need for D to be twist-reduced. Definition 3.1.
Let D be a reduced alternating surface link diagram on F . Fix a checker-board coloring on D . Let G A (resp. G B ) be the Tait graph (i.e., checkerboard graph)of D on F , whose edges correspond to crossings of D , and whose vertices correspond toshaded (resp. unshaded) regions of F − D , such that G A and G B are dual graphs on F .See Figure 1. Note that the Tait graph of a reduced alternating surface link diagram maycontain loops, but only homologically non-trivial ones. Let G (cid:48) A and G (cid:48) B be the reduced Taitgraphs obtained by deleting all but one edge in each set of parallel edges in G A and G B , asin Definition 2.2. A. CHAMPANERKAR AND I. KOFMAN
Figure 2.
Different kinds of cycles in the Tait graph are shown in differentcolors. From left to right: loop giving the diagram representativity 2 (cyan),nugatory crossing (green), null-homologous 2-cycle (pink), genus-generatingloops with representativity 4 (blue and red).
Figure 3.
Two alternating link diagrams are projected on F , partly shown.In both cases, the red crossing and the blue crossing are homologically twist-equivalent. One Tait graph has homologous loops (left) or a null-homologous2–cycle (right). Neither pair of crossings forms a twist region on F .See Figure 2 for several examples of different kinds of cycles in the Tait graph on thesurface F . Definition 3.2.
Recall, two edges in G are parallel if they are homologous on F . Twocrossings of D are homologically twist-equivalent if their corresponding edges are parallel ineither G A or G B . The homological twist number τ F ( D ) is defined as the number of homo-logical twist-equivalence classes of crossings of D . Thus, each homological twist correspondsto one set of parallel edges in G A or G B , which is one edge in G (cid:48) A or G (cid:48) B .See Figure 3 for two examples of homologically twist-equivalent crossings of D on F ,which do not form a twist region on F . Proposition 3.3. If t F ( D ) denotes the twist number, as in [10, Definition 2.4] , of a stronglyreduced, twist-reduced WGA diagram, then τ F ( D ) ≤ t F ( D ) ≤ τ F ( D ) . Moreover, if g ( F ) ≤ or the representativity r ( D, F ) ≥ , then τ F ( D ) = t F ( D ) .Proof. Let G A and G B be the Tait graphs of D on F , which do not contain loops since D is strongly reduced. A pair of edges in G A or G B is parallel if and only if they forma null-homologous 2–cycle. If it bounds a disk ∆ on F , then the hypothesis that D istwist-reduced, as in [10, Definition 2.5], implies that ∆ or a disk in F − ∆ contains a twistregion of D , which is the same as a homological twist-equivalence class of crossings of D .Thus, the two definitions of twist number agree in this case. VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 9
Figure 4.
For D on the torus (left), states s A (middle) and s B (right) areshown. Here, | s A | = 2 , | s B | = 1 , r ( s A ) = r ( s B ) = 1 , k ( s A ) = 1 , k ( s B ) = 0.On the other hand, suppose the null-homologous 2–cycle bounds a subsurface F (cid:48) ⊂ F which is not a disk, so it forms an essential separating curve on F . Hyperbolicity precludesboth vertices from being 2–valent, but if one vertex is 2–valent, then D has a bigon on F and the two crossings are homologically twist-equivalent. So the two definitions of twistnumber agree in this case as well.However, if neither vertex is 2–valent, then the two crossings are homologically twist-equivalent, but are not part of a twist region because D is twist-reduced. Moreover, thisdiscrepancy occurs for every essential null-homologous 2–cycle without 2–valent vertices in G A or G B . This proves the inequality.Finally, an essential null-homologous 2–cycle in G A or G B bounds a compressing disk of F , and intersects the diagram D in 4 points. If g ( F ) ≤ r ( D, F ) ≥
5, then neither G A nor G B admit such a 2–cycle. In the remaining cases, τ F ( D ) = t F ( D ). (cid:3) The Jones-Krushkal polynomial
In [11], Krushkal defined a homological Kauffman bracket derived from his 4-variablepolynomial p G ( x, y, u, v ), and proved the invariance of a two-variable generalization of theJones polynomial for links in thickened surfaces. We will use a later variant J K ( t, z ), calledthe reduced Jones-Krushkal polynomial, which was introduced by Boden and Karimi [3].Following [11], it is proved in [3] that J K ( t, z ) is an invariant of oriented links under isotopyand diffeomorphism of the thickened surface.We briefly recall the homological Kauffman bracket due to Krushkal [11]. Let F be aclosed orientable surface of genus g . Let K be a link in F × I , with a link diagram D on F . Suppose that D has c crossings, each of which can be resolved by an A –smoothing or B –smoothing. A state s of D is a collection of simple closed curves on F that results fromsmoothing each crossing of D . See Figure 4. Let a ( s ) and b ( s ) be the number of A and B –smoothings, and let | s | be the number of closed curves in s . Let s A and s B denote theall– A and all– B states of D , so that for the Tait graphs G A and G B , we have | V ( G A ) | = | s A | and | V ( G B ) | = | s B | . Let n = | V ( G A ) | − N = | V ( G B ) | −
1. Define k ( s ) = dim(kernel( i ∗ : H ( s ) → H ( F ))) ,r ( s ) = dim(image( i ∗ : H ( s ) → H ( F ))) , where i : s → F is the inclusion map. We call r ( s ) the homological rank of s , so that k ( s ) + r ( s ) = | s | . The homological Kauffman bracket is defined as follows: (cid:104) D (cid:105) F = (cid:88) s A ( a ( s ) − b ( s )) ( − A − − A ) k ( s ) z r ( s ) . To recover the usual Kauffman bracket for a classical diagram D , we set z = − A − − A anddivide by one factor of − A − − A . To obtain the Jones-Krushkal polynomial, which was the original link invariant defined in [11], we normalize by the writhe as usual, ( − A ) − w ( D ) (cid:104) D (cid:105) F ,and set A = t − / .If D is checkerboard colorable, then [ K ] = 0 in H ( F × I ) by [3], so it follows that k ( s ) ≥ s of D . So we can instead use the following version of the Jones-Krushkalpolynomial due to Boden and Karimi: Definition 4.1 ([3]) . Let K be an oriented link in F × I , represented by a checkerboard-colorable link diagram D on F . The reduced Jones-Krushkal polynomial is defined by J K ( t, z ) = ( − w ( D ) t w ( D ) / (cid:88) s t ( b ( s ) − a ( s )) / ( − t − / − t / ) ( k ( s ) − z r ( s ) . The reduced Jones-Krushkal polynomial specializes to the usual Jones polynomial V K ( t )by setting z = − t − / − t / . Any classical diagram will have r ( s ) = 0 for all states, so that J K ( t, z ) = V K ( t ) for every classical link K . However, there exist alternating virtual knotswith V K ( t ) = 1 but non-trivial J K ( t, z ).By [11, Theorem 6.1] for non-split D , we obtain (cid:104) D (cid:105) F from P G A ( X, Y, U, V ) as follows: (cid:104) D (cid:105) F ( A, z ) = A (2 g +2 n − c ) d z g P G A (cid:18) − A − , − A , A z , A z (cid:19) . With the additional normalization as in Definition 4.1, we obtain J K ( t, z ) by(7) J K ( t, z ) = ( − w t (3 w − g − n + c ) / z g P G A (cid:18) − t, − t − , z √ t , √ tz (cid:19) . Recall the definition of genus-generating loops and 3 –petal loops from Definition 2.4.
Definition 4.2.
For a reduced alternating diagram D on F , let (cid:96) ( G (cid:48) A ) and (cid:96) ( G (cid:48) B ) be thesubgraphs of loops in the reduced Tait graphs G (cid:48) A and G (cid:48) B . Define γ ( D ) = { { e , e } ⊂ (cid:96) ( G (cid:48) A ) | g ( H ( e ∪ e )) > } , ¯ γ ( D ) = { { e , e } ⊂ (cid:96) ( G (cid:48) B ) | g ( H ( e ∪ e )) > } . Theorem 4.3.
For a closed orientable surface F of genus g ≥ , let K be a non-splitoriented link in F × I that admits a reduced alternating diagram D on F , such that neitherof its Tait graphs has –petal loops. Let λ = | (cid:96) ( G (cid:48) A ) | , ¯ λ = | (cid:96) ( G (cid:48) B ) | , µ = b ( G (cid:48) A − (cid:96) ( G (cid:48) A )) , ¯ µ = b ( G (cid:48) B − (cid:96) ( G (cid:48) B )) , γ = γ ( D ) , ¯ γ = ¯ γ ( D ) . Then (8) τ F ( D ) = b ( G (cid:48) A ) + b ( G (cid:48) B ) − g = λ + µ + ¯ λ + ¯ µ − g and the reduced Jones-Krushkal polynomial J K ( t, z ) has the following coefficients: (9) ( − ( w + n ) t w +2 n + c (cid:16) ( − c t ( g − c ) (cid:16) ¯ λzt − (¯ µ − ¯ γ ) t (cid:17) − ( µ − γ ) t − + λzt − (cid:17) , where c and w are the crossing number and writhe of D , and n = | V ( G A ) | − . We prove Theorem 4.3 after the following corollary, which is important for Theorem 1.1.Recall that D is strongly reduced when neither G A nor G B has loops, so in particular, γ ( D ) = ¯ γ ( D ) = 0. In addition, γ ( D ) = ¯ γ ( D ) = 0 implies that neither Tait graph of D has3–petal loops. Corollary 4.4. If D is a reduced alternating diagram on F , such that γ ( D ) = ¯ γ ( D ) = 0 ,then τ F ( D ) is a link invariant of K in F × I . VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 11
Proof.
For g ( F ) = 0, the twist number is a link invariant by the proof of the Tait flypingconjecture in [14], so we may assume g ( F ) ≥
1. By [3], J K ( t, z ) is an invariant of K in F × I . Thus, by Theorem 4.3, τ F ( D ) is a link invariant when γ ( D ) = ¯ γ ( D ) = 0, and theterms in (9) are distinct terms in J K ( t, z ).The terms in (9) coincide when ( − c t ( g − c ) = ± t − or ± t − ; i.e., when c = g + 1 or c = g + 2. As D is cellularly embedded, c = | V A | + | V B | + 2 g − | V A | , | V B | ≥
1, whichallows only the cases: ( g, c ) ∈ { (1 , , (1 , , (2 , } . Moreover, both c = g + 1 and c = g + 2imply that either | V A | = 1 or | V B | = 1. So one Tait graph G consists of only loops, and as D is reduced alternating, these loops are homologically non-trivial.Let H ⊂ G . For [ ∂ H ] in H ( F ), let Λ( H ) = dim([ ∂ H ]). By [11, Equation (5.5)], g ( H ) + g ( F − H ) + Λ( H ) = g ( F ) . Since D is cellularly embedded, then so is G . Thus, for H = G consisting of homologicallynon-trivial loops, we have g ( F − H ) = Λ( H ) = 0. Hence, g ( H ) >
0, which implies thatat least one pair of loops in G must be genus-generating loops, which are excluded by thecondition γ ( D ) = ¯ γ ( D ) = 0.Thus, when γ ( D ) = ¯ γ ( D ) = 0, the terms in (9) are distinct terms in J K ( t, z ). (cid:3) The proof of Corollary 4.4 relies on the condition γ ( D ) = ¯ γ ( D ) = 0, but it may not benecessary. Question 4.5. If D is a reduced alternating diagram on F , is τ F ( D ) a link invariant of K in F × I ? Proof of Theorem 4.3. If g = 0, then D is a classical link diagram. In this case, λ = ¯ λ =0 since loops in its Tait graph can only come from nugatory crossings, so γ = ¯ γ = 0. Forclassical links, J K ( t, z ) = V K ( t ), so now both (8) and (9) follow from [7].To prove (8) for g >
0, we extend the argument in [7] to links in thickened surfaces. Let G A = ( V A , E A ) , G (cid:48) A = ( V A , E (cid:48) A ) , G B = ( V B , E B ) , G (cid:48) B = ( V B , E (cid:48) B ). Since G A and G B aredual graphs on F , | E A | = | E B | and | V A | + | V B | = | E A | + 2 − g . The homological twistnumber τ F ( D ) counts sets of homologically twist-equivalent crossings, which we can countusing sets of parallel edges in G A and G B , as follows: τ F ( D ) = | E A | − ( | E A | − | E (cid:48) A | ) − ( | E B | − | E (cid:48) B | )= | E (cid:48) A | + | E (cid:48) B | − | E A | = | E (cid:48) A | + | E (cid:48) B | − ( | V A | + | V B | − g )= ( | E (cid:48) A | − | V A | + 1) + ( | E (cid:48) B | − | V B | + 1) − g = b ( G (cid:48) A ) + b ( G (cid:48) B ) − g = λ + µ + ¯ λ + ¯ µ − g. We now prove (9) for g >
0. Let P G A ( X, Y, U, V ) be as in Theorem 2.3, with G = G A . Byduality [11, Theorem 3.1], P G B ( X, Y, U, V ) = P G A ( Y, X, V, U ). Therefore, by Theorem 2.3, λ, ¯ λ, µ, ¯ µ are exactly the coefficients of the following terms of P G A ( X, Y, U, V ):(10) µ V g X n − + λ V g − X n + ¯ µ U g Y N − + ¯ λ U g − Y N , where n = | V A | − N = | V B | −
1. Using χ ( F ) = | V A | + | V B | − c , we have n + N = c − g . Let π ( X α Y β U i V j ) ∈ Z [ t ± / , z ] denote the term in J K ( t, z ) obtained from X α Y β U i V j by the substitutions in (7). We evaluate each term in (10): π ( V g X n − ) = ( − w t w − g − n + c z g (cid:16) √ tz (cid:17) g ( − t ) n − = ( − ( w + n ) t w +2 n + c ( − t − ) ,π ( V g − X n ) = ( − w t w − g − n + c z g (cid:16) √ tz (cid:17) g − ( − t ) n = ( − ( w + n ) t w +2 n + c ( zt − ) ,π ( U g Y N − ) = ( − w t w − g − n + c z g (cid:16) z √ t (cid:17) g ( − t ) g + n − c +1 = ( − ( w + n ) t w +2 n + c ( − c t ( g − c ) ( − t ) ,π ( U g − Y N ) = ( − w t w − g − n + c z g (cid:16) z √ t (cid:17) g − ( − t ) g + n − c = ( − ( w + n ) t w +2 n + c ( − c t ( g − c ) ( zt ) . This verifies that the terms in (9) come from the corresponding terms in (10). We now findthe other terms in P G A ( X, Y, U, V ) that overlap with these terms in J K ( t, z ).For the µ –term, suppose π ( X α Y β U i V j ) = ± π ( V g X n − ). Since the RHS has no z factor,it follows that i + j = g . From exponents on t , we have α − β − i/ j/ g/ n − ⇒ α + j + 1 = β + g + n. If α = n − κ for some integer κ ≥
0, then n − κ + j + 1 = β + ( i + j ) + n ≥ ⇒ κ = 0 or κ = 1 . If α = n then β + i = 1, so β, i ∈ { , } . If α = n − β + i = 0, so β = i = 0. We areleft with only three possibilities: α = n − , β = 0 , i = 0 , j = g = ⇒ V g X n − α = n, β = 0 , i = 1 , j = g − ⇒ U V g − X n α = n, β = 1 , i = 0 , j = g = ⇒ V g X n Y We already know µ V g X n − is in P G A ( X, Y, U, V ). Since G A does not have 3–petal loops, wecan apply Lemma 2.5 to see that U V g − X n has coefficient γ in P G A ( X, Y, U, V ). As a term in J K ( t, z ), π ( V g X n − ) = − π ( U V g − X n ) because X n and X n − contribute opposite signs, sowe call it the ( µ − γ )–term in J K ( t, z ). For the final case above, we claim that V g X n Y cannotbe a term in P G A ( X, Y, U, V ). Suppose there exists H ⊂ G A whose weight contributes to V g X n Y . As in the proof of Lemma 2.5, the factor X n implies H ⊂ (cid:96) ( G A ). Because D is reduced alternating on F , all loops in G A are homologically non-trivial. The factor Y implies that H has weight with a factor y k for k >
0, so H must contain 3–petal loops,which are excluded by hypothesis. Thus, V g X n Y cannot be a term in P G A ( X, Y, U, V ).With the cases exhausted, we see that no other terms in P G A ( X, Y, U, V ) besides V g X n − and U V g − X n contribute to the ( µ − γ )–term in J K ( t, z ).For the ¯ µ –term, we can use duality [11, Theorem 3.1]: P G A ( X, Y, U, V ) = P G B ( Y, X, V, U ).If π ( X α Y β U i V j ) = ± π ( U g Y N − ), the argument above for the dual graph G B again impliesonly three possibilities: α = 0 , β = N − , i = g, j = 0 = ⇒ U g Y N − α = 0 , β = N, i = g − , j = 1 = ⇒ U g − V Y N α = 1 , β = N, i = g, j = 0 = ⇒ U g XY N By the same arguments on the dual graph, for D reduced alternating, only U g Y N − and U g − V Y N are terms in P G A ( X, Y, U, V ). Therefore, no other terms in P G A ( X, Y, U, V )besides these terms contribute to the (¯ µ − ¯ γ )–term in J K ( t, z ). VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 13
For the λ –term, suppose π ( X α Y β U i V j ) = ± π ( V g − X n ). Since the RHS has a z factor,it follows that i + j = g −
1. From exponents on t , we have α − β − i/ j/ g − / n = ⇒ α = β + i + n. If α = n − κ for some integer κ ≥
0, then n − κ = β + i + n ≥ ⇒ β = i = κ = 0 . This leaves only one possibility: α = n, β = 0 , i = 0 , j = g − ⇒ V g − X n . Therefore, no other terms in P G A ( X, Y, U, V ) besides V g − X n contribute to the λ –term in J K ( t, z ). For the ¯ λ –term, we can use a similar argument or use duality again.This completes the proof of (9). (cid:3) Lemma 4.6.
For K in F × I as in Theorem 4.3, only the terms V g X n and U g Y N of P G A ( X, Y, U, V ) contribute the extremal terms of J K ( t, , which has span ( c − g ) .Proof. By [3, Theorem 2.9], and dividing by one factor of − A − − A for the reducedpolynomial, the span of J K ( t,
1) is exactly ( c − g ). We now identify the subgraphs of G A that contribute the two extremal terms of J K ( t, P G A ( X, Y, U, V )which contributes the highest t –degree term of J K ( t,
1) has the highest X –degree and highest V –degree. Namely, the unique spanning subgraph H in G A with an empty edge set hasweight v g x n . Similarly, H = G A has weight u g y N , which contributes the the lowest t –degreeterm of J K ( t, P G A ( X, Y, U, V ) has the terms V g X n and U g Y N , which contributethe extremal terms of J K ( t, P G A ( X, Y, U, V ) contribute the extremal terms of J K ( t, H ⊂ G A whose weight also contributes to V g X n . Asin the proof of Lemma 2.5, the factor X n implies H has weight with factor x n and H ⊂ (cid:96) ( G A ). Thus, H has weight v g x n y k for k ≥
0. Because D is reduced alternatingon F , all loops in G A are homologically non-trivial. If k > H must contain 3–petal loops, which are excluded by hypothesis. Thus, only H contributes the term V g X n in P G A ( X, Y, U, V ). The argument for H = G A follows by duality [11, Theorem 3.1], P G B ( X, Y, U, V ) = P G A ( Y, X, V, U ). (cid:3) Proof of Theorem 1.1.
Since D is strongly reduced, λ = ¯ λ = 0. Thus, by (8), τ F ( D ) = µ + ¯ µ − g, which is a link invariant of K in F × I by Corollary 4.4.We claim that the µ and ¯ µ terms in (9) with λ = ¯ λ = 0 and γ = ¯ γ = 0 are exactlythe sub-extremal terms of J K ( t, V g X n and U g Y N of P G A ( X, Y, U, V ) contribute the extremal terms of J K ( t, c − g ). The µ and ¯ µ terms in (10) differ from V g X n and U g Y N , and in J K ( t,
1) they have span ( c − g − J K ( t, µ and ¯ µ terms have a z factor in J K ( t, z ). Thus, the µ and ¯ µ terms in (9)are exactly the sub-extremal terms of J K ( t, τ F ( D ) ≤ t F ( D ) ≤ τ F ( D ) . The volume bounds in Theorem 1.1 now follow from [10, Theorem 1.4]. Since essentialnull-homologous cycles occur only for g ≥
2, the bounds for g = 1 are the same. Since Figure 5.
The 2 × D with τ F ( D ) = 4, and self-dual Tait graphs G A = G B shown as a ribbongraph. τ F ( D ) ≤ t F ( D ), the lower bound for g ≥ t F ( D ) ≤ τ F ( D ), the upperbound for g ≥ (cid:3) Examples
Below we provide data to confirm Theorem 2.3 and Theorem 4.3 for three virtual links.
Example 1.
The 4–component virtual link K shown in Figure 5 is also discussed in [3,Example 3.10]. For its 2 × D on the torus, τ F ( D ) = 4. We havethe following data from this diagram: g = 1 , µ = 3 , λ = 0 , γ = 0 , ¯ µ = 3 , ¯ λ = 0 , ¯ γ = 0 , c = 4 , w = − , n = 1 , N = 1 . Eqn (8) : τ F ( D ) = λ + µ + ¯ λ + ¯ µ − g = 4Eqn (10) : µ V g X n − + λ V g − X n + ¯ µ U g Y N − + ¯ λ U g − Y N = 3 V + 3 UP G A ( X, Y, U, V ) =
V X + 6 +
U Y + 3 V + 3 U Eqn (9) : − ¯ λ ( zt − ) + (¯ µ − ¯ γ ) t − / + ( µ − γ ) t − / − λ ( zt − ) = 0 + 3 t − / + 3 t − / + 0 J K ( t, z ) = − t − / + 3 t − / + 3 t − / − t − / + (cid:0) zt − (cid:1) These results agree with Theorem 2.3 and Theorem 4.3. Note that λ = ¯ λ = 0, so we cancompute τ F ( K ) directly from the sub-extremal coefficients of J K ( t, T × I − K is 4 v oct , which is within the bounds of Theorem 1.1for τ F ( K ) = 4, although D has representativity 2. VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 15
Figure 6.
First row, left to right: Virtual knot 4.106, its diagram D on thetorus with τ F ( D ) = 3, and its Tait graphs G A (red) and G B (blue) on thetorus. Second row, left to right, shown as ribbon graphs: Tait graph G B andits reduction G (cid:48) B (blue), and G A (red) which is already reduced. Note thepair of genus-generating loops in G A . Example 2.
The virtual knot K = 4 .
106 is shown in Figure 6, with a diagram D shownon the torus. We have the following data from this diagram: g = 1 , µ = 1 , λ = 2 , γ = 1 , ¯ µ = 1 , ¯ λ = 1 , ¯ γ = 0 , c = 4 , w = − , n = 1 , N = 1 . Eqn (8) : τ F ( D ) = λ + µ + ¯ λ + ¯ µ − g = 3Eqn (10) : µ V g X n − + λ V g − X n + ¯ µ U g Y N − + ¯ λ U g − Y N = V + 2 X + U + YP G A ( X, Y, U, V ) =
U X + U Y + V X + V + 2 X + U + Y + 2Eqn (9) : ¯ λ ( − zt − / ) + (¯ µ − ¯ γ ) t − + ( µ − γ ) t − + λ ( − zt − / ) = − zt − / + t − + 0 − zt − / J K ( t, z ) = − t − + t − − (cid:16) − zt − / + 2 zt − / − zt − / (cid:17) These results agree with Theorem 2.3 and Theorem 4.3. Note that one of the coefficientsin (9) is zero because µ = γ = 1. In this case, τ F ( K ) cannot be computed directly fromthe coefficients of J K ( t, z ). Also, note that if we set z = − t − / − t / , then J K ( t, z ) = 1,so the virtual knot 4 .
106 has trivial Jones polynomial.
Figure 7.
First row, left to right: Virtual knot 4.105, and its diagram D on the torus with τ F ( D ) = 2. Second row, left to right, shown as ribbongraphs: Tait graph G B and its reduction G (cid:48) B (blue), and G A (red) which isalready reduced. Note the pair of genus-generating loops in G (cid:48) B . Example 3.
The virtual knot K = 4 .
105 is shown in Figure 7, with a diagram D shownon the torus. From the diagram on the torus, we can see τ F ( D ) = 2, but it is less apparentfrom the virtual link diagram which evokes the knot 8 . We have the following data fromthis diagram: g = 1 , µ = 2 , λ = 0 , γ = 0 , ¯ µ = 0 , ¯ λ = 2 , ¯ γ = 1 , c = 4 , w = − , n = 2 , N = 0 . Eqn (8) : τ F ( D ) = λ + µ + ¯ λ + ¯ µ − g = 2Eqn (10) : µ V g X n − + λ V g − X n + ¯ µ U g Y N − + ¯ λ U g − Y N = 2 V X + 2 P G A ( X, Y, U, V ) =
V X + U + V + 2 X + 2 V X + 2Eqn (9) : ¯ λ ( zt − / ) − (¯ µ − ¯ γ ) t − − ( µ − γ ) t − + λ ( zt − / ) = 2 zt − / + t − − t − + 0 J K ( t, z ) = t − + t − − t − + t − + (cid:16) zt − / − zt − / (cid:17) These results agree with Theorem 2.3 and Theorem 4.3. Note that because ¯ γ = 1 , τ F ( K )cannot be computed directly from the coefficients of J K ( t, z ). VOLUMISH THEOREM FOR ALTERNATING VIRTUAL LINKS 17
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Department of Mathematics, College of Staten Island & The Graduate Center, City Uni-versity of New York, New York, NY
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