Unoriented Virtual Khovanov Homology
UUNORIENTED VIRTUAL KHOVANOV HOMOLOGY
SCOTT BALDRIDGE, LOUIS H. KAUFFMAN, AND BEN MCCARTY
Abstract.
The Jones polynomial and Khovanov homology of a classical link are in-variants that depend upon an initial choice of orientation for the link. In this paper,we define and verify a Khovanov homology theory for unoriented virtual links. (Vir-tual links include all classical links.) The graded Euler characteristic of this homologyis proportional to a new unoriented Jones polynomial for virtual links, which is an in-variant in the category of virtual links. The unoriented Jones polynomial continues tosatisfy an important property of the usual (oriented) Jones polynomial: for classical oreven links, the unoriented Jones polynomial evaluated at one is two to the power of thenumber of components of the link. As part of extending the main results of this paperto non-classical virtual links, a new, simpler definition for integral Khovanov homologyis described. Finally, we define unoriented Lee homology theory for virtual links basedupon the unoriented version of Khovanov homology. The Khovanov homology theory forvirtual links defined here is a considerable simplification of the previous definitions dueto Manturov and to Dye, Kaestner and Kauffman. Introduction
In generalizing Khovanov homology to virtual links, Manturov and then Dye, Kaestnerand Kauffman [5, 28] created definitions of the Khovanov complex that depended heavilyupon the orientation of the link. By starting with an oriented virtual link, the orientationnecessitated the introduction of many auxiliary concepts not found in classical link theory:source-sink orientations, cut loci, and local and global orders. These had to be carefullychosen to simultaneously work together to get a well-defined Khovanov complex whosedifferential anti-commuted. These choices, in turn, made integral Khovanov homology forvirtual links hard to use both in theory and in practice. For example, it is practicallyimpossible to write computer code to calculate the Khovanov homology of virtual linksusing these definitions–there is a certain amount of human judgment needed to set up thecomplex for computation. Due to the difficulties baked into these definitions, mathemati-cians have been searching for a cleaner, simpler definition of the Khovanov complex forvirtual links for the past decade.One of the main results of this paper is a vastly simplified definition of Khovanovhomology for virtual links: We give a definition of the Khovanov complex equivalent tothat of Manturov and Dye, Kaestner, Kauffman, but this time we begin with an unoriented link. Recall that the Kauffman bracket polynomial does not require a choice of orientationfor the knot or link diagram to which it is applied. Using this as our starting point, wedefine the bracket homology of a diagram, which is the “categorification” of the Kauffmanbracket of that diagram. We show that this complex and differential for this homology hasa simple description that avoids the need for the complicating auxiliary choices. Since theKauffman bracket and bracket homology of two equivalent diagrams of the same virtual a r X i v : . [ m a t h . G T ] M a r SCOTT BALDRIDGE, LOUIS H. KAUFFMAN, AND BEN MCCARTY link differ by a grading shift, we can then normalize the complex by appropriate gradingshifts to get desired invariants of links.For oriented links, the orientation can be used to determine an overall grading shift inthe bracket homology to get the original oriented Khovanov homology (cf. Theorem 6 andTheorem 7). As an application of our new definition of Khovanov homology for virtuallinks, we introduce different grading shifts that also lead to an invariant homology foran unoriented link L . To describe these shifts, we need three numbers for any virtualdiagram: s + , s − , and m . The number m is simply the number of mixed-crossings of thediagram, i.e., the number of classical crossings between different components of the link.Each component can also have several self-crossings . We can bifurcate the self-crossingsinto two types, s + and s − , that correspond to the number of positive and negative self-crossings of the diagram (see Section 2 for definitions).Let ( C ( D ) , ∂ ) be the bracket complex defined in Section 4 for a diagram D of anunoriented link L . The unoriented Khovanov chain complex, ˜ C ( D ), is a gradings-shiftedversion of the bracket complex:˜ C ( D ) = C ( D )[ − s − − m ] { s + − s − − m } . Here the first grading shift is for the homological grading and the second is for the q -grading of the bi-graded complex. The homology of this complex, called unorientedKhovanov homology , is an invariant of the link (cf. Theorem 8): Theorem A.
Let L be an unoriented virtual link. The unoriented Khovanov homology,denoted (cid:103) Kh ( L ) , can be computed from any virtual diagram of L and is a virtual linkinvariant. The bracket homology and gradings-shift also leads to an unoriented version of Leehomology:
Theorem B.
Let L be an unoriented virtual link. The unoriented Lee Homology, Kh (cid:48) ( L ) ,is an invariant of the link L . This homology theory may have an impact on the study of the genera of unorientablespanning surfaces for both classical and virtual links. A priori there is no coherent way toorient the links appearing in a non-orientable cobordism. This renders the usual theoryimpractical for studying such cobordisms. Having a Lee homology theory that is inde-pendent of orientability removes this barrier.An immediate consequence of Theorem A (and Proposition 1) is that the graded Eulercharacteristic of the unoriented Khovanov homology defines a polynomial invariant thatdoes not depend on the orientation of the link: χ q ( (cid:103) Kh ( L )) = (cid:88) i,j ∈ Z ( − i q j dim( (cid:103) Kh i,j ( L )) . One might reasonably call this the “unoriented Jones polynomial” for the link (see J L inSection 2). However, this choice has two substantial deficiencies. First, the polynomial hascomplex, not real, coefficients. Second, even for classical links, evaluating the polynomial NORIENTED VIRTUAL KHOVANOV HOMOLOGY at q = 1 is not always positive as it is for the usual Jones polynomial. A significantamount of this paper is dedicated to correcting both of these deficiencies. It turns outthat the solution—cores and mantles—can be applied to fix other problems that havecome up in the virtual link theory literature.We can get a hint of how to fix these deficiencies by looking at classical links. Forclassical links, the required normalization is to multiply χ q ( (cid:103) Kh ( L )) by ( − λ where λ = (cid:80) i The unoriented Jones polynomial, defined by (cid:101) J L ( q ) = ( − ˜ λ − s − − m ) q ( s + − s − − m ) (cid:104) L (cid:105) , is an unoriented virtual link invariant. Moreover, (cid:101) J L ∈ Z [ q − , q ] . The definition of ˜ λ also solves the second deficiency. An even link is a virtual linkin which there are an even number of classical mixed-crossings for every component ina diagram of the link (cf. [15, 32]). Even links are sometimes called 2-colorable links inthe literature (cf. [36]). Since all virtual knots and classical links are even, one can oftengeneralize theorems of virtual knots and classical links to even virtual links. The nexttheorem is new in the literature for the usual (oriented) Jones polynomial for non-classical virtual links L . It also shows that ˜ λ correctly addresses the second issue for the unorientedJones polynomial (see Theorem 5 and Corollary 3): Theorem D. If L is an oriented virtual link with (cid:96) components, then (cid:101) J L (1) = J L (1) = (cid:40) (cid:96) if L is even if L is odd.The number, J L (1) , is independent of the choice of orientation. Thus, we prove that the usual (oriented) Jones polynomial, J L , counts the number ofcomponents of a virtual link when the link is even, extending this well-known property forclassical links, and that the unoriented Jones polynomial, (cid:101) J L , also preserves this property.In order to define (cid:101) λ , we needed to introduce a new idea in virtual link theory thatextends even link invariants to all virtual links: the identification of the core and mantle of a virtual link, and more generally, a multi-core decomposition (See Section 3.2). Amulti-core decomposition is the separation of a virtual link L into a set of invariant sub-links, L = C ∪ . . . ∪ C n ∪ M n , where each sub-link C i is even ( C is called the core), andthe sub-link M n is either the empty link, or it is odd ( M n is the final mantle).While there are a number of invariants for even virtual links in the literature, general-izing them to all virtual links has been elusive. They fail to be invariants for odd virtuallinks because the definition of the invariant often depends heavily on each componenthaving an even number of classical mixed-crossings. However, by identifying an invarianteven sub-link, the core, and eventually a set of invariant even sub-links in the multi-core SCOTT BALDRIDGE, LOUIS H. KAUFFMAN, AND BEN MCCARTY decomposition, one can derive an invariant of the (odd) link by applying the even linkinvariants to each even core in the decomposition (see Theorem 2). Theorem E. Any invariant of even links, Ψ , induces a tuple of invariants (Ψ( C ) , . . . , Ψ( C n )) for the multi-core decomposition of a virtual link L , and the tuple itself is an invariant of L . In the case of the unoriented Jones polynomial, the multi-core decomposition identi-fies a maximal set of maximal even sub-links on which the virtual link “acts like” aneven link. Thus, the unoriented Jones polynomial for odd links defined in this paper isthe polynomial that is the “closest to” the unoriented Jones polynomial for classical links. Remark. Before continuing with the contents of this paper, it is worth making someremarks about the theory of virtual knots and links. Virtual knot theory is the study ofembeddings of circles in three manifolds of the type M = S g × I where S g denotes an ori-entable surface of genus g and I denotes the unit interval. In other words, we examine theknot theory of three manifolds that are thickened surfaces. These embeddings are takenup to 1-handle stabilization of the manifolds M . The resulting theory has a diagrammaticinterpretation that generalizes the usual Reidemeister move description of classical knottheory, and indeed classical knot theory embeds in virtual knot theory. Many classicalinvariants generalize to virtual knot theory. For example, the Jones polynomial has ageneralization that is quite natural for the Kauffman bracket formulation, for example.One can regard virtual knot theory as a first step in understanding how to make directgeneralizations of the Jones polynomial to other three manifolds than the three dimen-sional sphere. (The well-known procedures using colored Jones polynomials and Kirbycalculus are indirect in the sense that the three manifold is represented via surgery onthe three-sphere.) One of the benefits of this direct definition of the Jones polynomial forvirtual links is that one can categorify it and study Khovanov homology for virtual linksin a way that is closely related to the original Khovanov complex. In this paper we showjust how close this association can be, by giving a definition of Khovanov homology forvirtual knots that is as close as possible to the classical definition. In [5], a generalizationof Rasmussen’s invariant was constructed for virtual knots and used to find the virtualfour-ball genus of positive virtual links. The relationship of virtual Khovanov homologyas a generalization of classical Khovanov homology was used to show that an infinite classof virtual knots with unit Jones polynomial are all non-classical. This result was obtainedby using the fact that Khovanov homology detects the classical unknot. These intimaterelationships between classical Khovanov homology and virtual Khovanov homology arenow made more intuitive by our work in this paper, and we expect that in subsequentwork the results of our new definitions will bear much fruit. Contents 1. Introduction 12. Unoriented Jones Polynomial for Classical Links 53. Unoriented Jones Polynomial for Virtual Links 7 NORIENTED VIRTUAL KHOVANOV HOMOLOGY Acknowledgements. Kauffman’s work was supported by the Laboratory of Topologyand Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Min-istry of Education and Science of the Russian Federation). All three authors would liketo thank William Rushworth for many helpful conversations and suggestions.2. Unoriented Jones Polynomial for Classical Links We introduce the main ideas of this paper by reviewing and generalizing the orientedversion of the Jones polynomial for classical links to the unoriented Jones polynomial.First, changing the orientation of a component of a link changes the usual (oriented)Jones polynomial in a controlled way. For example, if L = K ∪ K is a two componentlink with orientations chosen for K and K , let L (cid:48) be the same link but with the directionof K reversed so it has the opposite orientation. Then the Jones polynomials J L and J L (cid:48) in the variable A are related by(1) J L (cid:48) ( A ) = ( − A ) Lk ( K ,K ) J L ( A ) , where Lk ( K , K ) is the linking number of K and K with respect to the original orien-tations (cf. [33]). See Section 4 for definitions where we define J L ( A ) by the usual writhenormalization of the Kauffman bracket.It has been known for some time in the mathematical community how to adjust thenormalization of the original Kauffman bracket (in variable A ) to eliminate the factorin Equation (1). Since the original bracket is invariant under Reidemeister two andthree moves, one can limit the usual writhe normalization factor to the “self-writhe” ofcomponents of a link to compensate for Reidemeister one moves. Using the self-writheone can then write down a “Jones polynomial” in variable A that is invariant of theorientation as well. For a recent discussion of this fact, see [9]. In this paper, we usethe Kauffman bracket (in variable q ) that leads to the Khovanov homology construction.This requires adjusting the normalization factor in a different way.The new choice of normalization factor is based upon a few simple observations. Let L = K ∪ · · · ∪ K (cid:96) be a link with (cid:96) components and let D be a virtual diagram of L . A self-crossing in D is a classical crossing in which both arcs of the crossing are from the SCOTT BALDRIDGE, LOUIS H. KAUFFMAN, AND BEN MCCARTY same component. For a specific component K i , the usual signs for each self-crossing of K i are the same for either orientation of K i . Therefore we keep this information and let s + and s − be the total number of positive and negative self-crossings of a link L .A moment of reflection on the Reidemeister moves shows that the first Reidemeistermove is covered by the self-crossing data, and the third Reidemeister move should notplay a role in any normalization. For the second Reidemeister move, the case of using themove on a single component K i continues to be taken care of by the self-crossing data.The final case is a Reidemeister two move for strands from two different components ofa link. Therefore, the remaining type of crossing to consider is a mixed-crossing , that is,a classical crossing of a diagram involving two different components. Let m be the totalnumber of mixed-crossings of a diagram D . Note that we do not assign a positive ornegative sign to these crossings.Before addressing the final case, let us review how the Kauffman bracket changes forthis move. First, to get the Jones polynomial, one multiplies the Kauffman bracket by anormalization factor that depends on the writhe, i.e., J L ( q ) = ( − − n − q n + − n − (cid:104) L (cid:105) where n − and n + are the number of negative and positive (classical) crossings respectively. Here we are using the form of the Kauffman bracket given by (cid:104) (cid:105) = (cid:104) (cid:105) − q (cid:104) (cid:105) , and(2) (cid:104)(cid:13) ∪ D (cid:105) = ( q − + q ) (cid:104) D (cid:105) . (3)Clearly, the normalization depends upon the way the components are oriented, but theKauffman bracket itself does not require a choice of orientation (with the same thing hap-pening for bracket homology defined in Section 4 and Section 5). Applying the Kauffmanbracket to two strands with one over the other, we see that (cid:104) (cid:105) = − q (cid:104) (cid:105) . In other words, we can pull the strands apart at the cost of multiplying by an overallfactor of − q − (which partially motivates the usual normalization). Note that(4) ( − − m q − m , yields the proper correction factor. Further, note that the change in − m is equal to thechange in n + − n − under the move because, for any orientation of the strands, there willalways be one positive crossing and one negative crossing.Thus we may define the unoriented Jones polynomial of a classical link L by(5) ˜ J L ( q ) = ( − − n − q ( s + − s − − m ) (cid:104) L (cid:105) . The Jones polynomial is usually written with ( − n − instead of ( − − n − . These are equivalent since n − is an integer. Later in this paper, we work with half-integer powers of − − (cid:54) = ( − − . In this context, ( − − n − is the correct normalization. NORIENTED VIRTUAL KHOVANOV HOMOLOGY Based upon Expression (4), one should expect to see − s − − m as the exponent of − − n − . However, as we will elaborate in Section 3,(6) n − = − λ + s − + 12 m, where λ = (cid:80) i The unoriented Jones polynomial ˜ J L of a classical link L is an invariant ofthe link. Next we extend the definition of the unoriented Jones polynomial to virtual links. Thereare a number of pitfalls. The first is that m and Lk ( K i , K j ) can be half integers. Thismeans that the polynomial may have imaginary-valued coefficients. Since we wish topreserve (as much as possible) the well known fact that evaluating the Jones polynomialat 1 is 2 (cid:96) , this presents a problem. Worse yet, if the orientation of K i (or K j ) is reversed,then Lk ( K i , K j ) changes by an odd number if the total number of mixed-crossings between K i and K j is odd. Hence, the term λ needs to be modified to compensate for this issue.We tried many potential modifications that were orientation invariant, however, thesemodifications could not be normalized so that the well-known fact (2 (cid:96) ) continued to hold,even for classical links. The solution involves what we call the “core” of the virtual link,and it turns out that the core has far wider implications for virtual link theory.3. Unoriented Jones Polynomial for Virtual Links Before defining the core of a virtual link, we begin this section by recalling some basicfacts about virtual link theory.3.1. Virtual Knot Theory. Classical knot theory is the study of embeddings of disjointunions of S in S . Virtual knot theory, as introduced by Kauffman [18], is also the studyof disjoint unions of S , but in a different ambient space: Σ g × [0 , g denotesa closed orientable surface of genus g . Unlike links in many other 3-manifolds, virtuallinks have a diagrammatic theory akin to to that of classical links. We begin by recallingsome of the relevant facts about the theory, and refer the reader to [18, 20, 30] for a morethorough treatment.A virtual link diagram is a 4 − valent planar graph in which the vertices are decoratedwith classical crossings or virtual crossings , denoted by . Examples of virtual linkdiagrams are given in Figure 3 and Figure 10 .Just as two classical knots are equivalent if and only if their diagrams are relatedby a finite sequence of the Reidemeister moves shown in Figure 1, two virtual knots areequivalent if and only if their diagrams are related by a finite sequence of the Reidemeistermoves of Figure 1 together with the virtual Reidemeister moves in Figure 2. In particular,one may think of the virtual link itself as an equivalence class of virtual link diagrams,each member of which is related to the rest by a finite sequence of classical and virtualReidemeister moves. SCOTT BALDRIDGE, LOUIS H. KAUFFMAN, AND BEN MCCARTY R R R Figure 1. The classical Reidemeister moves. V1V2 V3VM Figure 2. The virtual Reidemeister moves.A classical link diagram is simply a virtual link diagram without virtual crossings.Thus, classical knot theory is a proper subset of virtual knot theory. For an in-depthtreatment of the diagrammatic theory of virtual links see [18].3.2. The Even Core of a Virtual Link. For classical links, the Jordan Curve Theoremimplies that each component of a link diagram intersects every other component of thediagram in an even number of crossings. However, virtual crossings are not genuinecrossings, which allows one to define useful notions of parity for virtual links. Definition 1 (Component-to-Component Parity) . Let D be a diagram of a virtual link L with components K , ..., K (cid:96) . The component-to-component parity , denoted π ( K i , K j ) ,is if there are an even number of mixed crossings between components K i and K j and otherwise. Definition 2 (Component Parity) . Let D be a diagram of a virtual link L with components K , ..., K (cid:96) . Define the parity of K i , denoted π ( K i ) , to be the number of mixed crossingsof D involving component K i , modulo 2. Remark 1. Let m i be the number of mixed classical crossings involving component K i ,and let v i be the number of virtual crossings involving K i and some other component. Bythe Jordan Curve Theorem, m i + v i ≡ (mod2) , Hence, the parity of K i is also the number of virtual crossings, mod 2, between K i and allother components. NORIENTED VIRTUAL KHOVANOV HOMOLOGY Definition 3 (Link Parity) . Let D be a diagram of a virtual link L with components K , ..., K (cid:96) . The link L is called even if all components of L are even, i.e., π ( K i ) = 0 forall i , and called odd if there exists an odd component. The parity of L , denoted π ( L ) , is π ( L ) = 0 if L is even and π ( L ) = 1 if L is odd. Note that the virtual Reidemeister moves have no effect on the parity, and the classicalReidemeister moves clearly leave the parity unchanged, and hence we obtain: Lemma 1. The link parity, each component parity, and each component-to-componentparity of a link are all virtual link invariants. It is well-known that even virtual links form a subset of virtual links for which itis often easier to define invariants (cf. [15, 32, 36]). However, odd virtual links presentcertain challenges. Some of these challenges may be overcome by choosing an invarianteven sublink. Definition 4. Given a diagram of a virtual link L = K ∪ K ∪ . . . ∪ K (cid:96) we obtain a sub-link L by deleting all the odd components of L . The resulting sub-link L may be evenor odd. If it is even, stop. Otherwise, repeat the procedure on L to get L , and continueuntil an even sub-link, L k , is obtained ( L k may be the empty link, which is even). Callthe even sub-link L k the core of L and denote it by C . Note that after deleting the odd components of L i to get L i +1 , it is possible thatcomponents of L i +1 that were even in L i become odd in L i +1 . However, Lemma 1 stillapplies to the sub-link L i +1 thought of as a link by itself. Therefore the deletion processis unique: L will always be even or odd as its own link, and the odd components of L that are deleted to get L will always be the same components, and so on. Thus, weimmediately obtain: Lemma 2. Any invariant of the even core of a virtual link L is an invariant of L . Invariants of odd links calculated from the even core leave out much of the informationabout the link. In order to recapture some of that information it is helpful to look at thecomplement of the even core, which we call the mantle of the virtual link. Definition 5. The mantle of a link L is the sub-link M given by the complement of thecore, i.e., M = L \ C . The astute reader will note that the mantle may be either an even or odd link in itsown right, and possesses its own even core. Hence, we may repeat the process describedabove. Given a virtual link L , determine its core C and mantle, M (note: L = M ∪ C )using Definition 4. Next, determine the core of M and denote it by C using Definition 4.This process also determines a new mantle, M . At this stage, L = C ∪ C ∪ M . Repeatthis process until M n is either empty or has an empty core (if M n is non-empty and hasempty core, it must contain only odd components). Thus, one obtains a decompositionof the link: L = C ∪ . . . ∪ C n ∪ M n . We call this the multi-core decomposition of the link.Repeated applications of Lemma 2 yield the following theorem. Theorem 2. Any invariant of even links, Ψ , induces a tuple of invariants (Ψ( C ) , . . . , Ψ( C n )) for the multi-core decomposition of a virtual link L , and the tuple itself is an invariant of L . In particular, this theorem implies that any invariant of even links immediately gen-eralizes to a new invariant of all virtual links—not just even links. For example, thepapers [32] and [15] generalize the odd writhe of a virtual knot [19] to even virtual links.In [15], a virtual orientation is described where the orientation of a component changesdirection in a diagram D at every virtual crossing (which is why the link must be even).Using this orientation, the sum of the signs of classical crossings between K i and K j whose over arc is K i is denoted Λ D ( i, j ). Clearly, this number depends upon the order:Λ D ( i, j ) (cid:54) = Λ D ( j, i ) in general. Corollary 1. [cf. Theorem 14 of [15], see also [32]] Let L = K ∪ K ∪ · · · ∪ K (cid:96) be anordered virtual link. Every core C r inherits an ordering from the ordering of L . For everycore C r of L , and any pair of components, K i and K j in C r , the number | Λ C r ( i, j ) | is aninvariant of the ordered unoriented virtual link L . Another generalization of the odd writhe to even links was given in [36]. In that paper,an even link is equivalent to the link being 2-colorable. The definition of 2-color writhe, J ( D ), of an even link diagram D depends upon a special parity function on each crossingthat satisfies the parity axioms (briefly mentioned below). For a given 2-coloring of anoriented link, define a quantity for that coloring as the sum of classical crossing signs ofonly the odd crossings (where odd is defined by that parity function). The 2-color writheof the link, J ( L ), is then a tuple of these numbers—one for each 2-coloring of a diagram D of L . This tuple is defined up to permutations of the entries. Corollary 2. The tuple consisting of the 2-color writhe of each core, ( J ( C ) , . . . , J ( C n )) ,is an invariant of the oriented virtual link L . Rushworth shows that for a virtual knot, J ( K ) is the odd writhe of K (see Proposition3.4 of [36]).Sometimes these tuples of invariants can be profitably added together to get an overallinvariant of the link. We want more, however. For an upgraded definition of λ for allvirtual links (not just even links), we wish to define “linking numbers” for all components,not just components in each of the cores. To do so, we introduce a parity function definedfrom pairs of components of L . This parity function is defined on components, but couldbe described as a parity function on crossings (cf. [31, 36]), as we will see next.3.3. A Parity Function. Let L = K ∪ . . . ∪ K (cid:96) be a virtual link with multi-coredecomposition, L = C ∪ C ∪ . . . ∪ C n ∪ M n . We define a component parity function on pairs of components as follows:(7) p ( K i , K j ) = (cid:26) K i , K j ∈ C r for some r p ( K i , K j ) = 0 we say that the two components link evenly, and if p ( K i , K j ) = 1 we say that the two components link oddly.Our component parity function descends to a parity function on crossings, as describedin [36] and [31]. If c is a crossing between component K i and K j , then we assign theparity, p ( K i , K j ), to c . Any self-crossing is assigned a parity of 0. Axioms 0 and 1(cf. [36]) are clearly satisfied. Because all crossings between K i and K j will have the sameparity, Axiom 2 is clearly satisfied. For a Reidemeister 3 move, we either have all three NORIENTED VIRTUAL KHOVANOV HOMOLOGY components in the same core, C r , two in the same core, and one is not, or, no two of thestrands are in the same core. These correspond to the three allowable cases of Axiom 3.We could just work with the parity function on crossings, but find it convenient to workwith the component parity function at the component-to-component level for reasons thatwill become apparent below.3.4. The Unoriented Jones Polynomial of a Virtual Link. We wish now to gener-alize the unoriented Jones polynomial (Equation (5)) for classical links to an invariant ofunoriented virtual links. As observed in Section 2, for a classical link, one may define theunoriented Jones polynomial of a link by(8) ˜ J L ( q ) = ( − λ − s − − m q s + − s − − m (cid:104) L (cid:105) , where λ = (cid:80) i 1) is unchanged by theorientation swap.For an odd virtual link, the unoriented Jones polynomial as defined for even/classicallinks need not be an orientation invariant, as the following example shows. Example 1. Let L and L be the oriented virtual Hopf links shown on the left and right,respectively in Figure 3. Observe that unoriented Jones polynomial defined for classicallinks is not orientation invariant, because λ − s − − m for L is − while for L it is . Figure 3. Virtual Hopf links with two different orientations.Thus, we need to extend the unoriented Jones polynomial defined for classical/evenlinks to odd links. This amounts to redefining the term λ . We do this next. Consider adiagram of an oriented virtual link L = K ∪ . . . ∪ K (cid:96) . We first define a modified linkingnumber : (cid:102) Lk ( K i , K j ) = ( − p ( K i ,K j ) · ( Lk ( K i ,K j )+ π ( K i ,K j ) ) Lk ( K i , K j ) , where p ( K i , K j ) is the component parity function and π ( K i , K j ) is the component-to-component parity. Suppose L has multi-core decomposition L = C ∪ . . . ∪ C n ∪ M n . Themodified linking number is, up to sign, the ordinary linking number, and if K i and K j belong to the same core, C r , it has the same sign as the ordinary linking number. If K i and K j belong to different cores, or if at least one of them belongs to the mantle, M n , thenthe modified linking number may have the opposite sign as the ordinary linking number. Lemma 3. The modified linking number, (cid:102) Lk ( K i , K j ) , is an element of Z and is anoriented virtual link invariant.Proof. Consider a diagram of an oriented virtual link L , and a multi-core decomposition L = C ∪ . . . ∪ C n ∪ M n . If K i and K j belong to the same core, then the modified linkingnumber is just the ordinary linking number, and must be an integer since each core is aneven sub-link.If K i and K j do not belong to the same core, but they do interact in an even numberof mixed-crossings, then (cid:102) Lk is still an integer. Otherwise, K i and K j interact in an oddnumber of mixed-crossings. Thus Lk ( K i , K j ) is a half-integer, and because π ( K i , K j ) = 1in this case, the exponent on − (cid:102) Lk will be an integer.The fact that (cid:102) Lk is an oriented virtual link invariant follows from the fact that theordinary linking number and parity are link invariants. (cid:3) We can use the modified linking number to extend λ from even links to all virtual links:(9) ˜ λ ( L ) = (cid:88) ≤ i The number ˜ λ is an oriented virtual link invariant. Remark 2. When L is even, L is the core. Thus, for an even link L , λ = ˜ λ and ˜ λ extends the definition of λ to odd links. The number ˜ λ is well-behaved under a change of orientation. Suppose K s ∈ C r . Let L s be the link L with the same orientations on the components except the orientation of K s reversed. Let K s denote the component K s with the opposite orientation. Subtracting˜ λ ( L ) − ˜ λ ( L s ), we pick up only the terms where the orientation changes. Thus:˜ λ ( L ) − ˜ λ ( L s ) = (cid:88) ≤ j ≤ (cid:96) (cid:16) (cid:102) Lk ( K s , K j ) − (cid:102) Lk ( K s , K j ) (cid:17) = (cid:88) ≤ j ≤ (cid:96) (cid:16) ( − p ( K s ,K j ) · ( Lk ( K s ,K j )+ π ( K s ,K j ) )+( − p ( K s ,K j ) · ( − Lk ( K s ,K j )+ π ( K s ,K j ) ) (cid:17) Lk ( K s , K j ) . For any j such that K j ∈ C r , the term in the sum above becomes 2 Lk ( K s , K j ). Itmay be that K s and K j interact in an odd number of classical crossings, in which case2 Lk ( K s , K j ) may be odd. However, if that happens, it must do so for an even number ofsuch j , because C r is an even sub-link.For any j such that K j / ∈ C r , the component parity function is given by p ( K s , K j ) = 1,and so the exponents come into play. It is still possible that K s and K j interact in aneven or odd number of classical crossings. If K s interacts with K j in an odd number NORIENTED VIRTUAL KHOVANOV HOMOLOGY of classical crossings, then π ( K s , K j ) = 1 and one of the exponents above will be evenand the other will be odd. Hence that term will contribute 0. Otherwise, π ( K s , K j ) = 0and the exponents will both be even or both be odd. Thus, that term will contribute ± Lk ( K s , K j ), which is an even number, since Lk ( K s , K j ) is an integer in this case.If K r ∈ M n , the argument is similar to the previous case. Hence, these observations,together with Lemma 4, prove: Lemma 5. If L s is the virtual link obtained from L = K ∪ . . . ∪ K (cid:96) by reversing theorientation on component K s then ˜ λ ( L ) and ˜ λ ( L s ) differ by an even integer. By definition ˜ λ is possibly a half-integer, and in many cases, an integer. But, in eithercase the previous lemma guarantees that ( − ˜ λ is invariant of the choice of orientation of L , since changing the orientation on a component changes ˜ λ by an even integer. (Hereand throughout the paper, we take ( − = i.) Theorem 3. The complex number ( − ˜ λ is invariant of the choice of orientation of L . We now have all of the ingredients in place to define the unoriented Jones polynomial for any virtual link L . Definition 6 (Unoriented Jones Polynomial) . The unoriented Jones polynomial for avirtual link L is (10) (cid:101) J L ( q ) = ( − ˜ λ − s − − m ) q ( s + − s − − m ) (cid:104) L (cid:105) . Note that this definition extends both the definition of the unoriented Jones polynomialfor classical links (cf. Equation (5) and Equation (6)) and even links by Remark 2. When L is an even virtual link (i.e. when L is its own even core), these two formulas are identical.As observed in Theorem 3, ( − ˜ λ need not be a real number, but it turns out that( − ˜ λ − m is. In particular, given a diagram of a virtual link L = K ∪ . . . ∪ K (cid:96) , observethat if π ( K i , K j ) = 1 then (cid:102) Lk ( K i , K j ) is a half-integer. If there is an odd number of suchpairs, then (cid:101) λ will be a half-integer as well. Otherwise, (cid:101) λ will be an integer. Similarly,in counting the total number of mixed crossings, if there is an odd number of pairsof components such that π ( K i , K j ) = 1, then there will be an odd number of mixedcrossings in the diagram. Hence, m will be a half-integer in this case, and will be aninteger otherwise. In either case, noting that s − ∈ Z , we obtain the following lemma. Lemma 6. For any virtual link L , ˜ λ − m ∈ Z , and hence, ( − ˜ λ − s − − m ∈ {− , } . Theorem 4. The unoriented Jones polynomial, (cid:101) J L , is an unoriented virtual link invari-ant. Moreover, (cid:101) J L ∈ Z [ q − , q ] .Proof. The Kauffman bracket clearly does not depend on orientation. The normaliza-tion factor q ( s + − s − − m ) depends only on self-crossings and the total number of mixed-crossings, neither of which change under an orientation switch, and ( s − + m ) is orientationinvariant for similar reasons. Thus, by Theorem 3, the first statement follows.The second statement follows from Lemma 6 and the fact that q ( s + − s − − m ) (cid:104) L (cid:105) hasinteger coefficients. (cid:3) Evaluating the Unoriented Jones Polynomial at . In this subsection, weexplain why the choices of ( − − n − and ( − ˜ λ − s − − m are important normalization factorsfor the oriented and unoriented Jones polynomials, J L and (cid:101) J L . Namely, we will show thatevaluating either polynomial at 1 is either 2 (cid:96) for an (cid:96) -component even link or 0 if the linkis odd. We start with the following definition and lemma. Definition 7. We define a numerical invariant for virtual links by evaluating the bracketpolynomial at 1: [ L ] = (cid:104) L (cid:105) (1) . Observe that J L (1) = ( − − n − [ L ] and that (cid:101) J L (1) = ( − ˜ λ − s − − m [ L ]. Lemma 7. The usual (oriented) Jones polynomial and unoriented Jones polynomial eval-uated at are each invariant under crossing changes. That is, if L and L are orientedvirtual links with diagrams that are identical except for a single crossing change, then J L (1) = J L (1) , and (cid:101) J L (1) = (cid:101) J L (1) . Proof. Consider (cid:101) J first. The effect of a crossing change on [ L ] is to multiply by − 1. Thusit suffices to show that (cid:16) ˜ λ − s − − m (cid:17) changes parity under a crossing change. Suppose c is the crossing to be changed. If c is a self-crossing, then ˜ λ and m are unaffected (sincethey only consider mixed crossings), and s − changes by ± c is a mixed crossing then s − and m remain unchanged. If c involves two componentsin the same core of L , then the crossing change results in a net change of 1 in the linkingnumber. If the crossing change is between two components that do not belong to thesame core, then we consider the effect of the crossing change on (cid:102) Lk ( K i , K j ) where K i and K j are the two components that cross at c. Note that when the crossing is switched, Lk ( K i , K j ) will change by 1 which is sufficient to change the parity of (cid:102) Lk ( K i , K j ) asdesired.The proof for J is simpler: changing a positive crossing to a negative (or vice-versa)clearly changes the parity of n − and hence, compensates for the sign change of [ L ]. (cid:3) Remark 3. One possible extension of λ we considered was to use the classical/even λ onlyon the crossings in the even core(s) and ignore all other crossings. However, Lemma 7shows why we needed a modified linking number that incorporated every mixed-crossing:Without a modified linking number calculated from every mixed-crossing, the lemma andtherefore main results of this subsection would not be true. It is well known that for a classical link the usual (oriented) Jones polynomial is, whenevaluated at 1, equal to two to the number of components of the link. It was not knownhow this result extended to virtual links. The following theorem is a new theorem in the(non-classical) virtual link literature. Theorem 5. If L is an oriented virtual link with (cid:96) components, then J L (1) = (cid:40) (cid:96) if L is even if L is odd. NORIENTED VIRTUAL KHOVANOV HOMOLOGY The number, J L (1) , is independent of the choice of orientation. We first noticed this result in the category of planar trivalent graphs (cf. [1, 4]), andwondered if a similar result held for virtual links. The fact that it does was an importantmotivation behind this current paper and [3] (See Future Aims, Section 8). Proof. Enumerate the components of L by L = K ∪ . . . ∪ K (cid:96) . The proof proceeds byinduction on the number of classical crossings in the diagram of L . The theorem is clearlytrue if there are no classical crossings.Suppose that there exists some crossing for a diagram of L . By repeated applicationsof Lemma 7 we can assume without loss of generality that every crossing of L is positive,i.e., n − = 0. Let c be a (positive) crossing between K i and some other component K j of L .If L A represents the link L with an A -smoothing at crossing c , and L B represents the link L with a B -smoothing at crossing c (see Section 4 for definitions of A - and B -smoothing),then(11) [ L ] = [ L A ] − [ L B ] . Both L A and L B have one less component than L since K i is welded to K j (see Figure 4). Figure 4. Smoothing a crossing.Observe that if c is oriented as shown on the left side of Figure 4, then the A -smoothing L A is compatible with this orientation, while the B -smoothing L B must be reoriented asshown in Figure 5. Let K ij stand for the component in L B with orientation given by K j and K i , i.e., the part of K i from L but with the opposite orientation. Figure 5. Reorienting L B .Since n − = 0 for L , J L (1) = [ L ]. Similarly, J L A (1) = [ L A ]. For L B , notice that if K i has k + 1 classical crossings in L , then after reversing the orientation of K i in L B to get anorientation for K ij , the welded K ij component will have k negative crossings in L B . Thus, J L B (1) = ( − k [ L B ]. Inserting these equations for L , L A , and L B into Equation (11), weget(12) J L (1) = J A (1) − ( − k J B (1) . Note that ( − k +1 = ( − π ( K i ) . Hence, we can rewrite Equation (12) as J L (1) = J A (1) + ( − π ( K i ) J B (1) . By induction, J A (1) must be either (2 (cid:96) − ) or 0 depending on the parity of L A . Similarly, J B (1) is either (2 (cid:96) − ) or 0 depending on the parity of L B . Moreover, since c was a crossingbetween different components of L , the parity of L A and L B are the same: π ( L A ) = π ( L B ).Thus, J L (1) = (cid:0) − π ( K i ) (cid:1) (cid:96) − · π ( L A ) . A similar argument applies when the chosen crossing involves only a single compo-nent. In that case, L A and L B will have an extra component, but the reasoning remainsessentially the same. Thus, by induction the theorem follows. (cid:3) The proof above is for the usual (oriented) Jones polynomial for virtual links, but theresult still holds for the unoriented Jones polynomial. There are two key observationsneeded to see why this is true. First, by Theorem 5, [ L ] = 0 if L is odd. Thus, we needonly check ˜ J L (1) for L even. Second, for an even virtual link, ˜ λ = λ (cf. Remark 2), and n − = − λ + s − + m . Putting these observations together, we get: Corollary 3. If L is an unoriented virtual link with (cid:96) components, then (cid:101) J L (1) = (cid:40) (cid:96) if L is even if L is odd. Thus, ˜ J L (1) = J L (1), which means ˜ J L continues to have the same non-negativity propo-erty when evaluated at 1 as the (oriented) Jones polynomial J L .Note that Corollary 3 could have been proven directly from Lemma 7, but the proofwould have obscured the equivalent arguments that are elucidated by the proof of Theo-rem 5.3.6. Other Component Parity Functions and Other Polynomials. There weretwo reasons for choosing the multi-core decomposition and the component parity functionabove. First, we wanted the polynomial invariant to be an invariant of the underlyinglink that does not depend on the orientation (cf. Theorem 4), and second, we wanted thepolynomial, when evaluated at q = 1 to count the number of components of the link (cf.Theorem 5 and Corollary 3). However, there are other component parity functions wemight have chosen to use in our definition of the modified linking number, each of whichmay be useful to other mathematicians in other contexts.For example, we could treat only pairs of components in the even core C = C of L aseven and ignore the other cores { C , . . . , C n } in the multi-core decomposition: p C ( K i , K j ) = (cid:26) K i , K j ∈ C NORIENTED VIRTUAL KHOVANOV HOMOLOGY Using this component parity function in the place of p in the definition of the modifiedlinking number leads to a different λ C and another polynomial J CL for which Theorem 4and Corollary 3 are both true. While this choice satisfies both of the criteria listed above,the component parity function p is still preferable, as it maximizes the behavior of evenlinks in L .Another option is to treat every pair of components as odd: p M ( K i , K j ) = 1 for all K i and K j . Using this parity to define a modified linking number has the virtue of giving rise to apolynomial, J ML , for which Theorem 4 is still true. However, Corollary 3 will fail to betrue.The last option we consider is in some sense the simplest, as it does not use λ , or itsvariants, at all. The polynomial we obtain from this choice is(13) J L ( q ) = ( − − s − − m ) q ( s + − s − − m ) (cid:104) L (cid:105) . The polynomial clearly does not depend on the orientation of the link, and hence is anunoriented virtual link invariant. However, it can have complex (not real) coefficients, soit does not satisfy the last part of Theorem 4, nor does it satisfy Corollary 3. Nevertheless, J L ( q ) has the advantage of being entirely determined by the graded Euler characteristicof the unoriented Khovanov homology described in Section 6, and unlike (cid:101) J L , J CL and J ML ,all of which require the choice of orientation of the link to define them (though they donot depend on that choice), J L does not require such a choice at all.4. Bracket Homology and Khovanov Homology In this section, we describe Khovanov homology along the lines of [6,23], and we tell thestory so that the gradings and the structure of the differential emerge in a natural way.This approach to motivating the Khovanov homology uses elements of Khovanov’s originalapproach, Viro’s use of enhanced states for the bracket polynomial [38], and Bar-Natan’semphasis on tangle cobordisms [7].We begin by working without using virtual crossings, and then we show how to introduceextra structure and generalize the Khovanov homology to virtual knots and links in thenext section.A key motivating idea involved in defining the Khovanov invariant is the notion ofcategorification. One would like to categorify a link polynomial such as the Kauffmanbracket (cid:104) D (cid:105) for a link diagram D of a link L . There are many meanings to the termcategorify, but here the quest is to find a way to express the link polynomial as a gradedEuler characteristic (cid:104) D (cid:105) = χ q ( H ( D )) for some homology theory associated with (cid:104) D (cid:105) . Remark 4. We will call this homology theory the bracket homology of D to emphasizethat this theory depends upon the diagram and is only categorifying the Kauffman bracketof that diagram. The bracket homology is an invariant of L up to some overall shift in thegradings. At the end of this section, we will show how to use an orientation of L to shiftthe gradings of the bracket homology to get the usual Khovanov homology invariant. The bracket polynomial [16, 17] model for the Jones polynomial [12–14, 39] is usuallydescribed by the inductive expansion of unoriented crossings into A -smoothings and B -smoothings on a link diagram D via:(14) (cid:104) (cid:105) = A (cid:104) (cid:105) + A − (cid:104) (cid:105) with(15) (cid:104) D (cid:13)(cid:105) = ( − A − A − ) (cid:104) D (cid:105) (16) (cid:104) (cid:105) = ( − A ) (cid:104) (cid:105) (17) (cid:104) (cid:105) = ( − A − ) (cid:104) (cid:105) Letting c ( D ) denote the number of crossings in the diagram D , if we replace (cid:104) D (cid:105) by A − c ( D ) (cid:104) D (cid:105) , and then replace A by − q − , the bracket can be rewritten in the the form ofEquation 2 and Equation 3: (cid:104) (cid:105) = (cid:104) (cid:105) − q (cid:104) (cid:105) with (cid:104)(cid:13)(cid:105) = ( q + q − ). It is useful to usethis form of the bracket state sum for the sake of the grading in the Khovanov homology(to be described below).We should further note that there is a well-known convention for describing the bracketstate expansion by enhanced states where an enhanced state has a label of 1 or x on eachof its component loops. We then regard the value of the loop q + q − as the sum of thevalue of a circle labeled with a 1 (the value is q ) added to the value of a circle labeledwith an x (the value is q − ) . We could have chosen the more neutral labels of +1 and − q +1 ⇐⇒ +1 ⇐⇒ q − ⇐⇒ − ⇐⇒ x, but, since an algebra involving 1 and x naturally appears later, we take this form oflabeling from the beginning.To see how the Khovanov grading arises, consider the form of the expansion of thisversion of the bracket polynomial in enhanced states. We have the formula as a sum overenhanced states s : (cid:104) D (cid:105) = (cid:88) s ( − n B ( s ) q j ( s ) where n B ( s ) is the number of B -type smoothings in s , r ( s ) is the number of loops in s labeled 1 minus the number of loops labeled x, and j ( s ) = n B ( s ) + r ( s ). This can berewritten in the following form: (cid:104) D (cid:105) = (cid:88) i ,j ( − i q j dim C i,j ( D )where we define C i,j ( D ) to be the linear span of the set of enhanced states with n B ( s ) = i and j ( s ) = j. Then the number of such states is dim C i,j ( D ) . We would like to turn the bigraded vector spaces C i,j into a bigraded complex ( C i,j , ∂ )with a differential ∂ : C i,j ( D ) −→ C i +1 ,j ( D ) . NORIENTED VIRTUAL KHOVANOV HOMOLOGY The differential should increase the homological grading i by 1 and preserve the quantumgrading j. Then we could write (cid:104) D (cid:105) = (cid:88) j q j (cid:88) i ( − i dim C i,j ( D ) = (cid:88) j q j χ (cid:0) C • ,j ( D ) (cid:1) , where χ ( C • ,j ( D )) is the Euler characteristic of the subcomplex C • ,j ( D ) for a fixed valueof j .This formula would constitute a categorification of the bracket polynomial. Below, weshall see how the original Khovanov differential ∂ is uniquely determined by the restrictionthat j ( ∂ s ) = j ( s ) for each enhanced state s . Since j is preserved by the differential, thesesubcomplexes C • ,j have their own Euler characteristics and homology. We have χ ( H • ,j ( D )) = χ ( C • ,j ( D ))where H • ,j ( D ) denotes the bracket homology of the complex C • ,j ( D ). We can write (cid:104) D (cid:105) = (cid:88) j q j χ ( H • ,j ( D )) . The last formula expresses the bracket polynomial as a graded Euler characteristic of ahomology theory associated with the enhanced states of the bracket state summation.This is the desired categorification of the bracket polynomial. Khovanov proved thata gradings-shifted version of this homology theory (using an orientation of the link) isan invariant of oriented knots and links, and that the graded Euler characteristic of thegradings-shifted version is the usual (oriented) Jones polynomial. Thus, he created a newand stronger invariant than the original Jones polynomial.The differential is based on regarding two states as adjacent if one differs from the otherby a single smoothing at some site. Thus, if ( s , τ ) denotes a pair consisting of an enhancedstate s and site τ of that state with τ of type A , then we consider all enhanced states s (cid:48) obtained from s by resmoothing at such a τ from A to B , and relabeling (with 1 or x )only those loops that are affected by the resmoothing. Call this set of enhanced states S (cid:48) [ s , τ ] . Then we shall define the partial differential ∂ τ ( s ) as a sum over certain elementsin S (cid:48) [ s , τ ] , and the differential for the complex by the formula ∂ ( s ) = (cid:88) τ ( − c ( s ,τ ) ∂ τ ( s )with the sum over all type A sites τ in s . Here c ( s , τ ) denotes the number of A -smoothingsprior to the A -smoothing in s that is designated by τ. Priority is defined by an initialchoice of order for the crossings in the knot or link diagram.In Figure 6, we indicate the original forms of the states for the bracket (not yet labeledby 1 or x to specify enhanced states) and their arrangement as a Khovanov categorywhere the generating morphisms are arrows from one state to another where the domainof the arrow has one more A -state than the target of that arrow. In this figure we haveassigned an order to the crossings of the knot, and so the reader can see from it how todefine the signs for each partial differential in the complex. A AAA A A A A AA A AB B BBB B B B BB BB1 23 Figure 6. Bracket states and Khovanov complex.We now explain how to define ∂ τ ( s ) so that j ( s ) is preserved. The unique form ofthe partial differential can be described by the following structure of multiplication andcomultiplication on the algebra V = k [ x ] / ( x ) where k = Z for integral coefficients:(1) The element 1 is a multiplicative unit and x = 0, and(2) ∆(1) = 1 ⊗ x + x ⊗ x ) = x ⊗ x. These rules describe the local relabeling process for loops in an enhanced state. Multiplica-tion corresponds to the case where two loops merge to a single loop, while comultiplicationcorresponds to the case where one loop bifurcates into two loops. It is easy to see that Proposition 1. The partial differentials ∂ τ ( s ) are uniquely determined by the conditionthat j ( s (cid:48) ) = j ( s ) for all s (cid:48) involved in the action of the partial differential on the enhancedstate s . The entire discussion above was for an unoriented link diagram D of a link L . Wewill return to such diagrams in the next section on virtual links, but before we do so, webriefly describe how to obtain the usual Khovanov homology from the bracket homology.Let { b } denote the degree shift operation that shifts the homogeneous component ofa graded vector space in dimension m up to dimension m + b . Similarly, let [ a ] denotethe homological shift operation on chain complexes that shifts the r th vector space in acomplex to the ( r + a )th place, with all the differential maps shifted accordingly (cf. [6]). NORIENTED VIRTUAL KHOVANOV HOMOLOGY Given an orientation of the link L , the crossings in the diagram D of L can be assignedto be positive or negative in the usual way. If n + and n − are the total number of positiveand negative crossings (classical, not virtual), we can shift the gradings of the brackethomology H ( D ) by [ − n − ] and { n + − n − } . Khovanov proved that this shifted brackethomology was an invariant of the oriented link: Theorem 6 (Khovanov, [23]) . Let D be a link diagram of an oriented link L . Then Kh ( L ) ∼ = H ( D )[ − n − ] { n + − n − } . It is this form of the theory that generalizes nicely to an unoriented version of Khovanovhomology (see Section 6).There is much more that can be said about the nature of the construction of thissection with respect to Frobenius algebras and tangle cobordisms. The partial boundariescan be conceptualized in terms of surface cobordisms. The equality of mixed partialscorresponds to topological equivalence of the corresponding surface cobordisms, and to therelationships between Frobenius algebras and the surface cobordism category. The proofof invariance of Khovanov homology with respect to the Reidemeister moves (respectinggrading changes) will not be given here. See [6, 7, 23]. It is remarkable that this versionof Khovanov homology is uniquely specified by natural ideas about adjacency of states inthe bracket polynomial.5. Bracket Homology of Virtual Links In this section, we describe how to define and calculate Khovanov homology for virtuallinks for arbitrary coefficients. We provide a new definition of integral Khovanov homologyfor virtual knots and links that is a much-desired simplification of the definitions of theKhovanov complex given previously in the literature. This new definition and algorithmprovides the same theory as earlier papers [28] and [5], but requires far less setup andterminology. It is an algorithm that is amenable for programming and we intend toimplement it and discuss computations in a sequel to this paper.Extending Khovanov homology to virtual knots for arbitrary coefficients is complicatedby the single cycle smoothing as depicted in Figure 7. Figure 7. Single cycle smoothing.We define a map for this smoothing η : V −→ V . In order to preserve the quantumgrading ( j ( s (cid:48) ) = j ( s ) via Proposition 1), we must have that η is the zero map. Consider the following complex in Figure 8 arising from the 2-crossing virtual unknot: η∆ η m Figure 8. Khovanov complex for the two-crossing virtual unknot.Composing along the top and right we have η ◦ η = 0 . But composing along the opposite sides we see m ◦ ∆(1) = m (1 ⊗ x + x ⊗ 1) = x + x = 2 x. Hence the complex does not naturally commute or anti-commute.When the base ring is Z / Z the definition of Khovanov homology given in the previoussection goes through unchanged. Manturov [28] (see also [29], [30]) introduced a defini-tion of Khovanov homology for (oriented) virtual knots with arbitrary coefficients. Dye,Kaestner and Kauffman [5] reformulated Manturov’s definition and gave applications ofthis theory. In particular, they found a generalization of the Rasmussen invariant andproved that virtual links with all positive crossings have a generalized four-ball genusequal to the genus of the virtual Seifert spanning surface. The method used in these pa-pers to create an integral version of Khovanov homology involves using the properties ofthe virtual diagram to make corrections in the local boundary maps so that the individualsquares commute (or anti-commute).Here we give a simplified version that obtains the same theory by starting with unori-ented diagrams. This definition is much simpler than the definitions given in [28] and [5].These papers started with oriented states for the Khovanov homology following patternsthat began in analysis of the Khovanov-Rozansky sl ( n ) homology theory [10, 24]. An in-dication of the simplicity of our definition is that Manturov’s original definition requiredGrassmann algebra structure in the Khovanov complex. This disappears in the new defi-nition. The original formulation of Khovanov homology as in [23] uses unoriented states. NORIENTED VIRTUAL KHOVANOV HOMOLOGY It took some time to realize that the construction for integral Khovanov homology can bedone in this unoriented and simpler manner. This formulation of the bracket homologyfirst occurs in the work of Kauffman and Ogasa (see the paper in preparation [21]).The simplified definition proceeds as follows. We set a base point on each loop of eachstate as indicated by the black dots in Figure 9. ∆ η η Figure 9. Transport across virtual crossings.Algebra to be processed by the local boundary maps is placed initially at these base-points. Then it is transferred to the site where the map occurs (either joining two loopsat that site, or splitting one loop into two at that site). Taking a path along the diagramfrom this basepoint to the site, one will pass either an even number of virtual crossings oran odd number. If the parity is even, then both x and 1 transport to x and 1 respectively.If the parity is odd, then x is transported to − x and 1 is transported to 1. The localboundary map is performed on the algebra as transported to the site, and then in theimage state, the result is transported back to the base point(s). This transport schemamodifies the local boundary maps in the complex so that all squares commute. We thendefine the signs for the full boundary maps exactly as we have done in the standardintegral Khovanov homology.In Figure 9 we have illustrated the situation where the top left state is labeled with 1and in the left vertical column we have ∆(1) = 1 ⊗ x + x ⊗ . We show how the initialelement 1 appears at the base point of the upper left state and how it is transported (as a1) to the site for the co-multiplication. The result of the co-multiplication is 1 ⊗ x + x ⊗ next composition of maps. In the Figure we illustrate the transport just for x ⊗ . Atthe new site this is transformed to ( − x ) ⊗ . Notice that the x in this transport movesthrough a single virtual crossing. We leave it for the reader to see that the transport of1 ⊗ x has even parity for both elements of the tensor product. Thus 1 ⊗ x + x ⊗ ⊗ x + ( − x ) ⊗ m (1 ⊗ x + ( − x ) ⊗ 1) = x − x = 0 . Thus we now have that the composition of the left sidesof the square is equal to the given zero composition of the right hand side of the square(which is a composition to two zero single cycle maps). Note that in applying transportfor composed maps we can transport directly from one site to another without going backto the base point and then to the second site.To get the virtual Khovanov homology of [5, 28], we calculate the bracket homology H ( D ) from a diagram D of a link L using the modifications described in this section.Next, we orient the link and use the total number of positive crossings n + and negativecrossings n − to shift the gradings of the bracket homology. Like the last theorem of thelast section, this homology theory is invariant of the oriented link L . It does not dependupon the diagram D . Theorem 7 (Manturov [28], Dye, Kaestner and Kauffman [5]) . For a diagram D of anoriented virtual link L , Kh ( L ) ∼ = H ( D )[ − n − ] { n + − n − } . Other definitions for virtual Khovanov homology have been given by [37] and [35]. Eachof these definitions give different solutions to handling the difficult morphism square dis-cussed above. 6. Unoriented Khovanov Homology We are now ready to describe the unoriented Khovanov homology in terms of thebracket homology and a grading shift (see Theorem 6). Let L be a virtual link and D adiagram of the link. The bracket homology H ( D ) is an invariant of the link up to gradingshifts, i.e., given any two diagrams D and D of the same link, there are numbers a and b such that H ( D ) ∼ = H ( D )[ a ] { b } . Recall that the bracket homology was defined withoutchoosing an orientation. The grading shifts ( − s − − m ) and ( s + − s − − m ) neededto define the unoriented Khovanov homology do not require the choice of an orientationeither (see the Introduction and Section 2).For a diagram D of a link L , shift the bracket chain complex ( C ( D ) , ∂ ) to get the unoriented Khovanov chain complex :(18) ˜ C ( D ) = C ( D )[ − s − − m ] { s + − s − − m } . (We do not include λ , as λ does require choosing an orientation and can jump by an eveninteger by choosing a different orientation.) Let (cid:103) Kh ( D ) be the homology of this chaincomplex. NORIENTED VIRTUAL KHOVANOV HOMOLOGY Theorem 8. Let L be an unoriented virtual link, and D and D be two diagrams of L thatare equivalent except they differ by one virtual Reidemeister move (virtual or classical).Then (cid:103) Kh ( D ) ∼ = (cid:103) Kh ( D ) .Proof. For a diagram D of a virtual link L , the homology of the unoriented Khovanovcomplex ˜ C ( D ) is equal to H ( D )[ − s − − m ] { s + − s − − m } . Since H ( D ) and H ( D ) areisomorphic up to a gradings shift, we only need to show that if (cid:103) Kh ( D ) ∼ = (cid:103) Kh ( D )[ a ] { b } ,then a = b = 0. Since D and D differ by a single Reidemeister move, we check each typeof move. Virtual moves V1, V2, V3 and VM (cf. Figure 2) do not effect the enhancedstates, s + , s − , or m . Hence (cid:103) Kh ( D ) = (cid:103) Kh ( D ) in that case. For the first classicalReidemeister move and the second classical Reidemeister move performed on the samecomponent, the terms − s − and s + − s − shift the bracket homology by the same numberas in the proof of the oriented Khovanov homology. For the second classical Reidemeistermove performed between two different components, H ( D ) ∼ = H ( D )[1] { } for the movethat removes two mixed-crossings in D . In this case, the term − m in both gradingshifts compensates for this change in grading. Finally, the third classical Reidemeistermove does not change s + , s − or m , and it induces is an isomorphism H ( D ) ∼ = H ( D ). Ineach case, a = b = 0, which was to be shown. (cid:3) A corollary of this theorem is that (cid:103) Kh ( D ) is isomorphic for every diagram D of L .Thus (cid:103) Kh ( D ) is an invariant of L which can be computed for any diagram D . Therefore,we can define: Definition 8. Let L be a unoriented virtual link and D be any virtual diagram of L .The unoriented Khovanov homology of L , denoted (cid:103) Kh ( L ) , is the homology of the complex ˜ C ( D ) . Remark 5. The gradings for the chain complex ( C ( D ) , ∂ ) are always integer valued, butthe gradings of the shifted unoriented Khovanov chain complex ( ˜ C ( D ) , ∂ ) may be half-integer valued. For classical links, the gradings of ˜ C ( D ) are always integer valued—there are alwaysan even number of mixed-crossings in a classical link, thus m is always an integer. Onemight conjecture that the same is true for even links since each component has an evennumber of mixed-crossings. This is not true, however, as the even virtualized Borromeanrings in Figure 10 shows: Figure 10. An even virtualized Borromean Rings with an odd number ofmixed-crossings.Since the homology (cid:103) Kh ( L ) can be graded by half-integers, we must extend the usualintegral grading to the additive group Z . The graded Euler characteristic for unorientedKhovanov homology is then χ q ( (cid:103) Kh ( L )) = (cid:88) i,j ∈ Z ( − i q j dim( (cid:103) Kh i,j ( L )) . Everything in the formula above continues to make sense if we choose the standardsquare root of − 1, i.e., ( − = i. The graded Euler characteristic of the unoriented Kho-vanov homology is a polynomial that may have imaginary valued coefficients. Therefore,evaluating the graded Euler characteristic at 1 is of the form i k · (cid:96) for an even virtual linkand 0 for an odd link. We could have defined the unoriented Jones polynomial as thisgraded Euler characteristic, which would yield J L (cf. Equation (13)). Maybe for somepurposes this would be the reasonable thing to do. However, normalizing the polynomialso that it evaluates to 2 (cid:96) (even) or 0 (odd) is desirable from the standpoint of matchingand generalizing already known theorems in classical link theory. The main motivationbehind working with the core and mantle of Section 3.2 was to establish an overall nor-malization that makes the unoriented Jones polynomial have integer-valued coefficientsand evaluates like the oriented Jones polynomial. It is the reason for the extra ( − ˜ λ inEquation (10). Thus, up to a well defined “sign,” the unoriented Khovanov homologycategorifies the unoriented Jones polynomial: Theorem 9. Let L be a virtual link. Then ˜ J L ( q ) = ( − ˜ λ χ q ( (cid:103) Kh ( L )) , where ˜ λ ∈ Z . The complex number ( − ˜ λ , i.e., the “sign,” is calculated by choosing an orientationof the virtual link diagram of L , but once computed, the result is independent of thechoice of that orientation by Theorem 3. Hence, while (cid:103) Kh ( L ) does not require choosingan orientation for L at any point in the construction, the unoriented Jones polynomialdoes. However, both (cid:103) Kh ( L ) and ˜ J L are invariant of that choice of orientation. NORIENTED VIRTUAL KHOVANOV HOMOLOGY Remark 6. If one is willing to temporarily orient the link (as we do to calculate ˜ J L ),the homological grading of ˜ C ( D ) could be changed so that there is no ( − ˜ λ factor inTheorem 9. Define a function ˜ l of ˜ λ to the set { , , , } by ˜ l = λ ≡ if 2˜ λ ≡ λ ≡ if 2˜ λ ≡ The value ˜ l is the same number for any orientation by Theorem 3. Replacing ( − s − − m ) with (˜ l − s − − m ) in Equation (18) gives an orientation-invariant Khovanov homologywhose graded Euler characteristic is ˜ J L . Lee Homology of Unoriented Links Lee [27] makes another homological invariant of knots and links by using a differentFrobenius algebra. She takes the algebra A = k [ x ] / ( x − 1) with x = 1 , ∆(1) = 1 ⊗ x + x ⊗ , ∆( x ) = x ⊗ x + 1 ⊗ ,(cid:15) ( x ) = 1 ,(cid:15) (1) = 0 . This can be used to define a differential ∂ (cid:48) and a link homology theory that is distinctfrom Khovanov homology. In this theory, the quantum grading j is not preseved, but wedo have that(19) j ( ∂ ( α )) ≥ j ( α )for each chain α in the complex. This means that one can use j to filter the chain com-plex for the Lee homology. The result is a spectral sequence that starts from Khovanovhomology and converges to Lee homology.We can extend Lee’s Frobenius algebra to virtual links to get a bracket complex forLee theory as follows. The involution defined in Section 5 that takes x (cid:55)→ − x as itis transported through a virtual crossing leads to a well-defined bracket chain complex,( C (cid:48) ( D ) , ∂ (cid:48) ), for the algebra A . This complex is filtered in the sense of Equation (19).After shifting overall by the unoriented gradings-shifts presented in this paper, we get aLee theory for a link that does not require a choice of orientation to define the homology: Theorem 10. Let L be an unoriented virtual link and D be any virtual diagram of L .The unoriented Lee Homology Kh (cid:48) ( L ) , i.e., the homology of the chain complex ( C (cid:48) ( D )[ − s − − m ] { s + − s − − m } , ∂ (cid:48) ) , is an invariant of the link L . The usual (oriented) Lee homology is simple for classical links. One has that thedimension of the Lee homology is equal to 2 (cid:96) where (cid:96) is the number of components of thelink L (cf. Theorem D!). Up to homotopy, Lee’s homology has a vanishing differential,and the complex behaves well under link concondance. In his paper [8], Dror BarNatanremarks, “In a beautiful article Eun Soo Lee introduced a second differential on theKhovanov complex of a knot (or link) and showed that the resulting (double) complexhas non-interesting homology. This is a very interesting result.” Rasmussen [34] usesLee’s result to define invariants of links that give lower bounds for the four-ball genus,and determine it for torus knots. Rasmussen’s invariant gives an (elementary) proof of aconjecture of Milnor that had been previously shown using gauge theory by Kronheimerand Mrowka [25, 26].In [5], Lee homology was generalized to virtual knots and links. Applications of it tounoriented links can be articulated again with the methods of the present paper. We willcarry this out in detail in our next paper [2].8. Future Aims This paper has been devoted to formulating an unoriented version of the Jones poly-nomial (via a normalization of the Kauffman bracket polynomial) and a correspondingversion of Khovanov homology for virtual knots and links that is based in unorientedlink diagrams. The resulting Khovanov homology is a reformulation of the Manturov [28]version of Khovanov homology and it improves on the methods of Dye, Kaestner andKauffman [5]. We intend the present paper as a basis for further research and wish tomake the following points about future work. • The dependence of the invariant on a choice of orientations is useful in certain con-texts. For example, orientations are useful in the context of oriented cobordisms.An invariant of the underlying link is useful as well and may inform on unorientedcobordisms. Our next paper explores the unoriented version of Lee homology forvirtual links described above and its applications to cobordisms, genus, and Ras-mussen invariants [2]. • This paper grew out of a search for an invariant in a different context: the 2-factorpolynomial for ribbon graphs. A ribbon graph G with a perfect matching M canbe made to behave like a knot by orienting the cycles in G \ M (see [1]). However,to define an invariant of a ribbon graph that is independent of the choice of perfectmatchings of the graph and orientations on the complementary cycles required an“orientation free” invariant. We will describe this new invariant of ribbon graphsin [3]. • There is a computer program for virtual homology as formulated by Tubben-hauer ( ) and a computer program forKhovanov homology for classical links available at Dror Bar Natan’s website NORIENTED VIRTUAL KHOVANOV HOMOLOGY ( http://katlas.org/wiki/Khovanov_Homology ). We plan to generalize the Math-ematica program of Bar Natan to the version of Khovanov homology for virtualknots and links that is expressed in the this paper.At the present time, we know remarkably little about virtual Khovanov homology. It isour intent that this situation will begin to change with the tools developed in this paper. References [1] S. Baldridge. A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings. Preprint,arXiv/1810.07302, 2018.[2] S. Baldridge, L. H. Kauffman, and B. McCarty. Applications of Unoriented Khovanov and LeeHomology. Preprint.[3] S. Baldridge, L. H. Kauffman, W. Rushworth. New invariants of ribbon graphs. Preprint.[4] S. Baldridge, A. Lowrance, and B. McCarty. The 2-Factor Polynomial Detects Even PerfectMatchings. Preprint, arXiv:1812.10346, 2018.[5] H. A. Dye, A. Kaestner, L. H. Kauffman. Khovanov homology, Lee homology and a Rasmusseninvariant for virtual knots. J. Knot Theory Ramifications 26 (2017), no. 3, 1741001, 57 pp.[6] D. 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Witten. Quantum Field Theory and the Jones Polynomial. Comm. in Math. Phys. (1989),351-399. Department of Mathematics, Louisiana State University, Baton Rouge, LA E-mail address : [email protected] Department of Mathematics, Statistics and Computer Science, 851 South MorganStreet, University of Illinois at Chicago, Chicago, Illinois 60607-7045Department of Mechanics and Mathematics, Novosibirsk State University, Novosi-birsk, Russia E-mail address : [email protected] Department of Mathematical Sciences, University of Memphis, Memphis, TN E-mail address ::