Computations of the slice genus of virtual knots
William Rushworth

TL;DR
This paper computes and estimates the slice genus of virtual knots with up to six crossings using extended Rasmussen invariants, advancing understanding of virtual knot concordance and genus calculations.
Contribution
It introduces new bounds for Rasmussen invariants in virtual knots and applies them to compute slice genus for knots with up to six crossings.
Findings
Computed slice genus for all virtual knots with 4 or fewer crossings.
Estimated slice genus for 46 virtual knots with 5 and 6 crossings.
Identified cases where two Rasmussen invariant extensions agree and proved additivity of one extension.
Abstract
A virtual knot is an equivalence class of embeddings of into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of into a thickened surface ; what is the minimal genus of oriented surfaces with the embedded as boundary, where is an oriented -manifold with ? We compute and estimate the slice genus of all virtual knots of classical crossings or less. We also compute or estimate the slice genus of virtual knots of and classical crossings whose slice status is not determined in the workâŠ
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Computations of the slice genus of virtual knots
William Rushworth
Department of Mathematical Sciences, Durham University, United Kingdom
This paper subsumes the (now withdrawn) arXiv submission On the virtual Rasmussen invariant.
Abstract.
A virtual knot is an equivalence class of embeddings of into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of into a thickened surface ; what is the minimal genus of oriented surfaces with the embedded as boundary, where is an oriented -manifold with ?
We compute and estimate the slice genus of all virtual knots of classical crossings or less. We also compute or estimate the slice genus of virtual knots of and classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum.
Key words and phrases:
Rasmussen invariant, virtual knot concordance, slice genus
1991 Mathematics Subject Classification:
57M25, 57M27, 57N70
1. Introduction
1.1. Statement of results
A virtual knot is an equivalence class of embeddings of into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and handle stabilisations whose attaching spheres do not intersect the embedded ; virtual links are defined analogously [17]. They are represented diagrammatically using knot diagrams with an extra crossing decoration, the virtual crossing , up to the virtual Reidemeister moves; see Figure 3 for such a diagram.
The slice genus of a virtual knot is defined in direct analogy to that of classical knots (see Section 1.2); it is less well-studied than that of classical knots, but obstructions to sliceness of virtual knots have been developed by a number of authors. They include the index polynomial of Heinrich [14] and the graded genus of Turaev [29]. Boden, Chrisman, and Gaudreau [4] have used these invariants and others to compute or estimate the slice genus of a very large number of the virtual knots of crossing or less (as given in Greenâs table [13]).
In another direction, Manturov and Fedoseev have produced slice obstructions for free knots [24, 10, 11]. A free knot is an equivalence class of -valent graphs, and a Gauss code representing a virtual knot may be projected to a code representing a free knot by forgetting the signs and directions of its chords. Given a free knot , obstructing the sliceness of necessarily obstructs the sliceness of every virtual knot which projects to it.
We shall focus on the Rasmussen invariant. It has been extended to virtual knots in two different ways, producing two distinct Rasmussen-like invariants: the virtual Rasmussen invariant due to Dye, Kaestner, and Kauffman [9], and the doubled Rasmussen invariant due to the author [27]. Both of these extensions provide obstructions to the sliceness of virtual knots (again see Section 1.2).
In this paper we employ these extensions of the Rasmussen invariant to compute or estimate the slice genus of virtual knots. The extensions themselves are derived from two distinct generalisations of Khovanov homology to virtual links, reviewed in Section 2. The results of the computations are given in two tables which begin on section 5 and section 5 respectively, and are outlined in Section 1.1.1.
Let be a virtual knot; postponing their definition until Section 2, let and denote, respectively, the virtual Rasmussen invariant, and the doubled Rasmussen invariant. Like the classical Rasmussen invariant the quantities and are difficult to compute, in general (in constrast can be computed by hand, as described below). In Section 4 four integer quantities are associated to a diagram of - , , , and - which allow for the estimation of and .
Theorem** (Theorems 4.4 and 4.19 of Section 4).**
Let be a diagram of a virtual knot . Then
[TABLE]
and
[TABLE]
The bounds and are generalisations of the slice-Bennequin bounds due, independently, to Kawamura [20] and [22] (see Section 1.3). They are easy to compute for any diagram of . Further, there are classes of diagrams for which the quantities and simplify. In fact, in Section 4.1.2, we characterise exactly the class of diagrams for which so that .
As an aside, we show that although the extensions and are distinct in general there is a class of virtual knots on which they agree.
Definition 1.1**.**
A classical crossing within a virtual knot diagram is even if it is resolved into its oriented resolution in the alternately colourable smoothing of ; otherwise it is odd. A virtual knot diagram is known as even if all of its classical crossings are even. A virtual knot is even if it possesses an even diagram. â
Remark**.**
This definition of odd and even crossings is shown to be equivalent to the standard definition involving Gauss codes in [27, Proposition ].
Classically, the oriented smoothing is necessarily alternately colourable (so that every classical knot is even). Virtually, this is no longer the case; consider the diagram given in Figure 3 (both of its classical crossings are odd).
Theorem** (Corollary 2.14 of Section 2).**
Let be an even virtual knot. Then .
As a final aside, we show that the virtual Rasmussen invariant is additive with respect to connect sum. By an abuse of notation denotes any of the knots which can be obtained as a connect sum between and .
Theorem** (Theorem 3.2 of Section 3).**
For virtual knots and
[TABLE]
1.1.1. Results of the computation of and
Section 5 contains two tables which give the results of the computation of the bounds and , along with the results of the computation or estimation of the slice genus which follows (see Section 1.4). The first table, beginning on section 5, contains the results for all virtual knots of classical crossings or less, as given in Greenâs table [13]. The second table, beginning on section 5, contains the results for of the virtual knots of classical crossings or less whose slice status is not determined in [5].
Many of the calculations and estimations of the virtual and doubled Rasmussen invariants are made by identifying that the knot in question is a connect sum, and applying the additivity of both invariants under that operation.
1.2. Virtual cobordism
In direct analogue to those of the classical case we make the following definitions (see [9] and [19]). Two virtual knot diagrams and are cobordant if one can be obtained from the other by a finite sequence of births and deaths of circles, oriented saddles, and virtual Reidemeister moves. Such a sequence describes a compact, oriented surface, , such that . If we say that and are concordant. If is the unknot, and is concordant to we say that is slice. In general, we define the slice genus of a virtual knot , denoted , as
[TABLE]
(here we have simply capped off the unknot in with a disc). It is natural to ask whether or not the slice genus of a classical knot may be lowered by treating it as a virtual knot. That is, given a classical knot, does the addition of virtual Redeimeister moves allow one to construct a surface bounding it of lower genus than its classical slice genus? This has been answered in the negative by Boden and Nagel [6], a concordance analogue to the result of Goussarov, Polyak, and Viro that classical links are left unaltered if one views them as virtual links [12].
Behind the scenes, the cobordism surface is embedded in a -manifold of the form , where is a compact, oriented -manifold with , where denotes a closed oriented surface of genus . The -manifold is described in the standard way in terms of codimension submanifolds and critical points: starting from , codimension submanifolds are until we pass a critical point, after which they are . Critical points of correspond to handle stabilisation. A finite number of handle stabilisations are needed to reach .
As mentioned in the abstract the slice genus of a virtual knot may be defined in a more natural manner. Let be a virtual knot and (by an abuse of notation) let be representative of . Then
[TABLE]
That this second definition is equivalent to the first follows from the observation that given two representatives of in and with , there exists a cylinder (embedded in a thickened oriented -manifold) which cobounds them. Further, this definition highlights the higher-dimensional topology at play when one considers the slice genus of virtual knots. In constrast to the classical case, in which the slice genus of a knot depends only on how surfaces bounding that knot may be embedded into , the slice genus of a virtual knot depends on the surface and on the -manifold .
1.3. The slice-Bennequin bounds
The Rasmussen invariant of a classical knot extracts geometric information from Khovanov homology, yielding a lower bound on the slice genus [26]. Given a classical knot it is, in principle, difficult to compute its Rasmussen invariant, denoted , as it is equivalent to the maximal filtration grading of all elements homologous to a certain generator of the Lee homology of .
Kawamura [20] and Lobb [22] independently defined diagram-dependent upper bounds on , denoted (for a diagram of ), which are easily computable by hand, along with an error term, , the vanishing of which implies that , in fact. More precisely,
[TABLE]
The bounds are henceforth referred to as the strong slice-Bennequin bounds; in Section 4 we construct analogous bounds on the virtual and doubled Rasmussen invariants.
1.4. Estimating the slice genus
This paper is concerned with the computation of the slice genus of virtual knots. These computations are achieved using the obstructions to sliceness offered by the two extensions of the Rasmussen invariant mentioned above. As stated, the virtual Rasmussen invariant, and one component of the doubled Rasmussen invariant are difficult to compute (this necessitates the construction of the bounds as mentioned in Section 1.1). The other component of the doubled Rasmussen invariant is, however, readily computable. Precisely, the quantity can be computed from quickly from any diagram of , as it is equal to the odd writhe of . That is:
Theorem** ([Proposition of [27]).**
Let be a diagram of a virtual knot . Let denote the sum of the signs of the odd crossings of . This is a knot invariant, known as the odd writhe of , and denoted [18]. Then .
Theorem** (Theorem of [27]).**
Let be a virtual knot such that . Then is not slice.
Whilst it is more difficult to compute, the other component of the doubled Rasmussen invariant also obstructs sliceness.
Theorem** (Corollary of [27]).**
Let be a virtual knot with . If then is not slice.
The virtual Rasmussen invariant provides a lower bound on the slice genus of a virtual knot.
Theorem** (Theorem of [9]).**
Let be a virtual knot. Then .
The computations and estimations of the slice genus are made as follows. Let be the diagram of a virtual knot given in Greenâs table [13], then:
- (i)
Compute , , , , and for , in order to estimate or compute and . 2. (ii)
Take the greatest of the upper bounds on provided by the estimations or computations of , , and . 3. (iii)
Attempt to find a cobordism from to the unknot of genus equal to the greatest upper bound on , thus computing . 4. (iv)
Failing that, find a cobordism of higher genus so that a region in which lies is identified.
1.5. Plan of the paper
First, in Section 2, we outline the issues faced when extending Khovanov homology to virtual links, and review two distinct ways of overcoming them i.e. two extensions of Khovanov homology to virtual links. Further, we review the extensions of the Rasmussen invariant produced from each of the homology theories. We also identify in Section 2.4 a class of virtual knots for which the two extensions of the Rasmussen invariant are equal.
Next, in Section 3, we produce canonical chain-level generators of one of the relevant homology theories. This is done by simplifying the decorated diagrammatic generators defined in [9], so that elements of the algebraic chain complex may be read off from them.
These canonical generators are required in Section 4, in which we construct the strong slice-Bennequin bounds on both the virtual and the doubled Rasmussen invariant. In this we follow much the same path as Lobb [22]; in fact, in the case of the virtual Rasmussen invariant, we recover formulae identical to his. In the case of the doubled Rasmussen invariant, however, the formulae arrived at are substantially different, a consequence of the structural differences between doubled Khovanov homology and its classical predecessor.
Finally, in Section 5, we use the tools we have developed to compute or estimate the slice genus of a large portion of the knots given in Greenâs table [13].
Acknowledgements. We thank A Referee for very helpful comments on an earlier version of this paper, and Hans Boden, Micah Chrisman, and Robin Gaudreau for sharing and discussing their work.
2. Review
We review the two homology theories used throughout this work. In an attempt to avoid confusion we shall refer to the theory due to Manuturov and reforumulated by Dye, Kaestner, and Kauffman as MDKK homology, and denote it by . We denote the other theory in question, doubled Khovanov homology, by . Classical Khovanov homology, where required, is denoted by . The perturbed versions of the theories are denoted by , , and .
The review of MDKK homology contained in Section 2.2 is substantially more detailed than the review of doubled Khovanov homology (contained in Section 2.3). This is because the methods used in Section 4 require chain-level generators of the complexes and . We are already in possession of such generators in case of but not . (In Section 3 we construct these generators.)
Before outlining the homology theories we describe the complications one encounters when attempting to extend Khovanov homology to virtual links.
2.1. Extending Khovanov homology
Manturov first defined Khovanov homology for virtual links [23]. His theory was reformulated by Dye, Kaestner, and Kauffman in order to define a virtual Rasmussen invariant [9]. An alternative extension of Khovanov homology to virtual links is doubled Khovanov homology, which provides the doubled Rasmussen invariant [27]. Here we briefly outline the problems encountered in attempting to extend Khovanov homology to virtual links, and the paths taken in [9] and [27] to overcome them.
The fundamental obstruction to transferring Khovanov homology to the virtual setting is the existence of the single-cycle smoothing depicted in Figure 1(A) (otherwise known as a one-to-one bifurcation). If the module assigned to a circle within a smoothing is the same as that assigned by classical Khovanov homology the map associated to this smoothing, denoted , must be identically zero, in order to preserve the quantum grading. This, in turn, causes the face depicted in Figure 1(B) to fail to commute. Notice that the differential along the top and right-hand edges is , but along the left-hand and bottom edges it is so that .
Thus classical Khovanov homology must be augmented in order to detect this face, if one wishes to assign the zero map. This is the approach taken by Manutrov and subsequently Dye et al, and outlined in Section 2.2. In [27] another approach is taken: the module assigned to a circle within a smoothing is altered, allowing for to be assigned a non-zero map while being grading preserving. The resulting theory is outlined in Section 2.3.
Remark**.**
Tubbenhauer [28] has constructed a virtual Khovanov homology theory in the manner of Bar-Natan [2] using non-orientable cobordisms, but there are compatibility issues with the theory presented in [9].
2.2. Review of MDKK homology
We review the construction of MDKK homology and the virtual Rasmussen invariant.
2.2.1. The complex
Let for a commutative ring and . In order to detect the problem face a symmetry present in (which corresponds to the two possible orientations of ) is exploited using the following automorphism:
Definition 2.1**.**
The barring operator is the map
[TABLE]
Applying the barring operator is referred to as conjugation. â
Note that if and then and the barring operator is just standard complex conjugation. How the barring operator is applied within the Khovanov complex is determined using an extra decoration on link diagrams, the source-sink decoration as depicted in Figure 2. A new diagram is formed by replacing the classical crossings with the source-sink decoration, which induces an orientation on the incident arcs of a crossing. Arcs of the diagram on which the induced orientations due to separate crossings disagree are marked by a cut locus. We refer the reader to [9].
2.2.2. The virtual Rasmussen invariant
There is a degeneration of Khovanov homology due to Lee [21]. There is such a degeneration of MDKK homology also. Dye, Kaestner, and Kauffman use the methods of Bar-Natan and Morrison [3] to show this. Specifically, they employ the Karoubi envelope of a category and the interpretation of virtual links as abstract links [7, 15], and define the virtual Rasmussen invariant.
As such diagrams are used extensively below, we describe the process given in [15] to obtain a (representative of an) abstract link from a (representative of a) virtual link (examples are given in Section 3). Let be a diagram of a virtual link, as in Figure 3, then
- (i)
About the classical crossings place a disc as shown in Figure 4. 2. (ii)
About the virtual crossings place two discs as shown in Figure 5. 3. (iii)
Join up these discs with collars about the arcs of the diagram.
The result is a knot diagram on a surface which deformation retracts onto the underlying curve of the diagram. We will denote abstract link diagrams by for a knot diagram and a compact, oriented surface (which deformation retracts on to the underlying curve of ). We treat such diagrams up to stable equivalence, defined below.
Definition 2.2** (Definition 3.2 of [7]).**
Let and be abstract link diagrams. We say that and are equivalent, denoted , if there exists a closed, connected, oriented surface and orientation-preserving embeddings , such that and are related by Reidemeister moves on . We say that two abstract link diagrams and are stably equivalent if there is a chain of equivalences
[TABLE]
for . â
Stable equivalence classes of abstract link diagrams are in bijective correspondence to equivalence classes of virtual link diagrams [15].
Definition 2.3**.**
A smoothing of an abstract link diagram is a diagram formed by smoothing the crossings of into either their [math]- or -resolution on . The result is a collection of disjoint copies of on the surface . A copy of is called a cycle. â
The diagram-level canonical generators of the Lee complex given in [9] are smoothings of abstract link diagrams with extra information added. This extra information keeps track of the source-sink structure of the virtual knot. The information is in the form of cross cuts which are added in the following way: before beginning the procedure described above mark the virtual knot diagram with cut loci as inherited from the source-sink orientation and preserve them on the abstract link diagram. Replace each cut locus with a cross cut which bisects the surface as shown in Figure 6. Henceforth by abstract link diagram we mean an abstract link diagram with cross cuts.
Using the source-sink decoration we add yet more information to abstract link diagrams in the form of a checkerboard colouring:
Definition 2.4**.**
From an abstract link diagram form its associated checkerboard coloured abstract link diagram from the surface and curve pair (where denotes the source-sink diagram formed by replacing each crossing by the source-sink decoration) by colouring the surface using the recipe given in Figure 7 and Figure 8.
Notice that Figure 7 allows us to induce a checkerboard colouring of smoothings of abstract link diagrams by simply joining the shaded or unshaded areas produced by smoothing the crossing. â
From checkerboard coloured smoothings of abstract link diagrams we are able to produce the tools used by Dye, Kaestner, and Kauffman to prove theorems analogous to those in [3]. Henceforth we set and .
Definition 2.5**.**
Let be the basis for where
[TABLE]
On the level of diagrams, arcs of a smoothing are coloured red or green to denote which generator they are labelled with. â
The properties of and are listed in Lemma of [9]. The most important for our purposes is that and are conjugates with respect to the barring operator. That is
[TABLE]
Definition 2.6** (Analogue of Definition of [3]).**
An alternately coloured smoothing of an abstract link diagram is a smoothing for which the arcs have been coloured either red or green such that the arcs passing through each crossing neighbourhood are coloured different colours. At a cut locus the colouring of an arc switches. â
Using alternately coloured smoothings the following theorems are stated and proved:
Theorem 2.7** (Theorem 4.2 of [9]).**
Within the Karoubi envelope the Lee complex of a virtual link is homotopy equivalent to a complex with one generator for each alternately coloured smoothing of on an abstract link diagram with cross cuts and with vanishing differentials.
Theorem 2.8** (Theorem 4.3 of [9]).**
A virtual link with components has exactly alternately coloured smoothings on an abstract link diagram with cross cuts. These smoothings are in bijective correspondence with the orientations of .
In Section 3 we describe the bijective correspondence of Theorem 2.8, but we conclude this section by stating the definition of the virtual Rasmussen invariant and its properties.
Definition 2.9**.**
Let be a virtual knot diagram, and the associated Lee complex and Lee homology, respectively. Let be the grading on induced by on . Define
[TABLE]
The virtual Rasmussen invariant of is
[TABLE]
Proposition 2.10** (Parts of Proposition and Theorem of [9]).**
The virtual Rasmussen invariant satisfies the following
- (1)
. 2. (2)
, for the mirror image of : the diagram formed by switching all positive classical crossings to negative classical crossings and vice versa. 3. (3)
, where denotes the slice genus of .
Notice that the virtual Rasmussen invariant lacks the out-of-the-box additivity of its classical counterpart (a consequence of the ill-defined nature of the connect sum operation on virtual knots). In Section 3.1 we show, however, that the virtual invariant is indeed additive.
2.3. Doubled Khovanov homology
We review doubled Khovanov homology and the doubled Rasmussen invariant.
2.3.1. Construction
Doubled Khovanov homology provides an alternative extension of Khovanov homology to virtual links [27]. The problem face is dealt with by âdoubling upâ the module assigned to a smoothing; this allows the map assigned to the single-cycle smoothing to be non-zero.
A schematic picture of this âdoubling upâ process is given in Figure 9; the left hand complex depicts the situation when the module is assigned to a cycle within a smoothing. One sees immediately that the map must be zero if it is to be degree-preserving. This is path followed by Manturov and Dye et al, and outlined in the previous section. The right hand complex, however, depicts the situation arrived at if one assigns the module to a cyle, where and (the superscripts are u for âupperâ and l for âlowerâ). This allows for to be non-zero and degree preserving.
Given a virtual link diagram, , the complex is formed in the usual way: form the cube of resolutions of , then assign modules to the vertices and maps to the edges. The module assigned to a smoothing of cycles is . The maps constituting the differential are as follows. The and maps are effectively unchanged:
[TABLE]
(notice that they do not map between the upper and lower summands). The map associated to the single cycle smoothing as in Figure 1(A) is given by
[TABLE]
We denote by the homology of the complex , where is the link represented by . We refer the reader to [27].
2.3.2. The doubled Rasmussen invariant
As in classical Khovanov and MDKK theories there is a perturbation of doubled Khovanov homology produced by adding a term of degree to the differential. As in the other cases, this perturbation allows the definition of a concordance invariant. In this section we give the essentials we require for Section 4.2, for full details we refer the reader to [27, Section ].
Given a virtual link diagram, , let denote the complex with the chain spaces of but with altered differential. The homology of is an invariant of the link represented by , and is denoted (where is the link represented by ). The complex is refered to as the doubled Lee complex, and the homology as the doubled Lee homology.
The rank of doubled Lee homology of a link depends on the number of alternately coloured smoothings the link possesses - here we mean the usual notion of alternately coloured smoothing, rather than the augmented notion of alternately coloured smoothings on abstract link diagrams used in Section 2.2. Unlike classical links, virtual links may posesses no alternately coloured smoothings. (In fact, one of the purposes of the extra decoration applied to diagrams in the construction of MDKK homology is to ensure that the oriented smoothing of the augmented diagrams is always alternately colourable.)
Theorem 2.11** (Theorem of [27]).**
Given a virtual link
[TABLE]
Further, given a diagram of a virtual link , each alternately coloured smoothing, , (if any exist) defines two generators of , denoted and and known as an alternately coloured generators.
A virtual knot has two alternately coloured smoothings [27, Theorem ] so that its doubled Lee homology is of rank . The four generators of the homology lie in a single homological degree, and the quantum grading of any one of them determines that of the others [27, Lemma ]. Thus, for a virtual knot, , the information contained in is equivalent to a pair of integers.
Definition 2.12** (Definition of [27]).**
For a virtual knot the doubled Rasmussen invariant is denoted , where is equivalent to the highest non-trivial quantum degree of , and is the single non-trivial homological degree of .
The component is easy to compute from any diagram, , of : it is the height of the alternately coloured smoothings of . It is also equal to the odd writhe of (see [27, Proposition ]).
2.4. Even knots
To conclude this section we give a class of virtual knots for which the two extensions of the Rasmussen invariant are equal.
Recall the definition of an even virtual knot given in Section 1.1; here prove a fact about the cube of resolutions associated to even virtual knot diagrams.
Proposition 2.13**.**
Let be an even virtual knot diagram. Then and contain no maps.
Proof.
As is even it possesses a global source-sink orientation i.e. applying the source-sink decoration does not yield any cut loci. (In fact, possessing a global source-sink structure is equivalent to being even, but here we only need one direction.) To see this orient with either of itâs orientations (the usual notion of orientation, not source sink), and consider leaving a classical crossing of and returning to the arc proscribed by the usual orientation. One sees from Figure 2 that passing through a classical crossing reverses the source-sink orientation. As all classical crossings of are even, one passes through an even number of crossings between leaving and returning at the proscribed arc. Thus the source-sink orientation has been reversed an even number of times, yielding no overall change. This argument can be applied at every crossing to show that has a global source-sink orientation.
Next, notice that every smoothing of inherits an orientation from the global source-sink orientation of : looking again at Figure 2 one sees that both resolutions of the classical crossing inherit an orientation from the source-sink decoration. That the orientation inherited is consistent between distinct classical crossings of follows from that fact that has no cut loci.
Finally, we notice that if every smoothing of inherits a coherent orientation from the global source-sink orientation of then every cycle within a smoothing must look as in the left or center of Figure 10, as the configuration on the right is prohibited for reasons of (source-sink) orientation. But we see that the configurations on the left and center correspond to either a merge or a split, while the configuration on the right corresponds to the single-cycle smoothing. Thus no single-cycle smoothings can occur in the cube of resolutions of and we arrive at the desired result. â
Corollary 2.14**.**
Let be an even virtual knot. Then so that .
Proof.
Let be an even diagram of . Then both and contain no maps by Proposition 2.13. As and do not map between the shifted and unshifted summands of , the complex splits as the direct sum . â
3. Chain-level generators of
In [9] canonical generators are produced at a diagrammatic level i.e. they are alternately coloured smoothings of (checkerboard-coloured) abstract link diagrams. These generators are sufficient to prove Theorems 2.7 and 2.8. Below, we give a method to produce the corresponding chain-level generators of . Before doing so, however, it is instructive to recall the bijection of Theorem 2.8 between orientations of a virtual link and alternately coloured smoothings of the associated abstract link diagram as given in [9]. We use Figure 3 as an example.
- (i)
Given a virtual link diagram construct the checkerboard coloured abstract link diagram as in Definition 2.4. Note that for a virtual knot the checkerboard colouring is independent of the orientation, a consequence of the invariance of the source-sink decoration under rotations. See Figure 11. 2. (ii)
For a given orientation of form the corresponding oriented smoothing on the checkerboard coloured abstract link diagram as in Definition 2.3. See Figure 12. 3. (iii)
Place a clockwise orientation on the shaded regions of the oriented smoothing, which in turn induces a new orientation on the arcs of the smoothing. On each arc compare this orientation to that induced by . If these two orientations agree colour the arc red, if they disagree colour the arc green (as in Definition 2.5). See Figure 13.
At this stage we have produced alternately coloured smoothings on abstract link diagrams as in Definition 2.6. We need a way of reading off from these diagrams elements of (as the oriented resolution is always at height [math]), which will be the chain-level canonical generators of . We are unable to do so at this point as the cycles of the alternately coloured smoothings possess more than one colour. We now describe a process by which single coloured smoothings can be produced, and hence chain-level generators of .
Firstly, we utilise the stable equivalence relation given in Definition 2.2 to work with alternately coloured smoothings of abstract link diagrams for which the surface deformation retracts onto the curve of the smoothing, for example the abstract link diagrams given in Figure 14. We can always do this as the curve of the smoothing is simply a disjoint union of copies of . Note that the resulting smoothing (of a checkerboard coloured abstract link diagram) may not be connected.
Next, we interpret the cross cuts as half-twists with the parity of the twist ignored. That is
[TABLE]
The author learnt of this interpretation in the talks of Dye and of Kaestner during Special Session 35, âLow Dimensional Topology and Its Relationships with Physicsâ, of the 2015 AMS/EMS/SPM Joint Meeting.
Replacing cross cuts with appropriate half-twists we are able to view the surface of the smoothing (of a checkerboard coloured abstract link diagram) as a two-sided surface such that the curve of the smoothing appears on both sides. That cross cuts always come in pairs ensures that the surface has two sides. Importantly, on each side of the surface the curve of the smoothing is coloured exactly one colour. This is because passing a cross cut causes the arc to change to change colour (c.f. Definition 2.6), and to pass a cut locus is to pass onto the other side of the surface. (From this one can see that passing a cut locus, or equivalently moving on to the other side of the surface, is replicated in by applying the barring operator.)
In summary, we view alternately coloured smoothings (of checkerboard coloured abstract link diagrams) such as those in Figure 14 as two sided surfaces such that the curve of the smoothing is coloured exactly one colour on each side. At this point it is clear that in order to read off generators of from such alternately coloured smoothings we must make a choice of side (or sides, if the surface of the smoothing is disconnected) of the surface to read. Further, we must also ensure that this choice is the same for both the alternately coloured smoothings associated to and . We must have this as they are both coloured versions of the same smoothing of an abstract link diagram (the oriented smoothing) c.f. the left hand smoothing of Figure 12 with Figure 13. In effect we are making the choice on this uncoloured smoothing, which the alternately coloured smoothings then inherit.
With all this in mind, let us make a choice: given a virtual knot diagram with orientations and , let denote the oriented smoothing of the checkerboard coloured abstract link diagram associated to . On cancel an arbitrary pair of adjacent cross cuts against one another so that the strand they bound is removed. An example is given in Figure 15. This cancellation of cross cuts is simply âflippingâ the segment of the surface they bound so that the other side of the surface is shown. Continue cancelling available arbitrary pairs of cross cuts until all have been removed. In our interpretation, that the smoothing has no cross cuts means that we are looking at exactly one side of surface. Now return to part (iii) of the process given on page iii, and colour the cycles of the oriented smoothings associated to and as dictated there. Denote by and the resulting alternately coloured abstract link diagrams associated to and , respectively. That the cycles of and are coloured with opposite colours follows from the fact that their orientations are opposite but the checkerboard colouring of and is the same.
Examples of such single coloured smoothings are given in Figure 16 and Figure 17. In this case a choice of top and bottom is equivalent to picking either the two smoothings on the left of the Figures, or the two on the right.
After all that we are left with smoothings of abstract link diagrams the cycles of which are coloured with exactly one colour, either red or green. We form the canonical generators of , denoted for an orientation of , by taking the appropriate tensor product of and as dictated by the colours of the cycles. In this way we obtain two distinct algebraic generators.
We conclude by remarking that the invariant is independent of this choice of which side of the surface to read. Making another choice results in an application of the barring operator to one or more tensor factors of and , because if a cycle is coloured green on one side of the surface it must be coloured red on the other. But conjugation does not interact with the filtration, that is
[TABLE]
To conclude this section we prove a Lemma analogous to Lemma of Rasmussen [26] which we will use in both the following sections.
Lemma 3.1**.**
Let be the number of components of . There is a direct sum decomposition , where is generated by all states with -grading conguent to , and is generated by all states with -grading congruent to . If is an orientation on , then is contained in one of the two summands, and is contained in the other.
Proof.
The first statement follows exactly as in the classical case. Regarding the second statement, following [26] let be the map which acts by the identity on and multiplication by on . We claim that . To show this we define a new grading on with respect to which has grading and has grading . We have that and so that and , and the map
[TABLE]
(which applies the barring operator to all tensor factors) acts as the identity on elements with new grading congruent to and multiplication by on elements with new grading congruent to . The new grading differs from the -grading by an overall shift so that
[TABLE]
as in the classical case. â
A direct corollary of Lemma 3.1 is that is not of top filtered degree, that is:
[TABLE]
3.1. Additivity of the virtual Rasmussen invariant
We can use the chain-level generators of to show that the virtual Rasmussen invariant is additive with respect to connect sum, confirming that the virtual invariant behaves in the same way as its classical counterpart in this respect.
The connect sum operation on virtual knots is ill-defined. That is, the result of the operation depends on both the diagrams used and the site at which the sum is conducted. As a result there exist multiple inequivalent virtual knots which can be obtained as connect sums of a fixed pair of virtual knots. By an abuse of notation we shall denote by any of the knots obtained as a connect sum of virtual knots and .
Theorem 3.2**.**
For virtual knots and
[TABLE]
Proof.
With the chain-level generators in place, along with Lemma 3.1, the proof follows much the same path as that in [26]. For all connect sums there exists the map
[TABLE]
It sends a canonical generator of to a canonical generator of of the form where is a generator of for . As in the classical case, the map is of filtered degree and we obtain
[TABLE]
From this point the proof proceeds as in that of the analogous statement in [26]: utilising the fact that we are able to obtain from Equation 3.3 that
[TABLE]
as required. â
In light of Theorem 3.2 we see that the Rasmussen invariant cannot distinguish between connect sums of a fixed pair of virtual knots. In general it is not known, for and both (possibly inequivalent) connect sums of a fixed pair of virtual knots, if is concordant to . It is known, however, that neither the Jones polynomial [25] nor the Rasmussen invariant can distinguish them. This leads one to posit whether Khovanov homology can; in the case of connect sums of trivial diagrams it is shown in [27] that doubled Khovanov homology cannot.
4. Computable bounds
In this section we extend the strong slice-Bennequin bounds to the virtual and doubled Rasmussen invariants. The bounds are constructed, and cases in which they vanish partly or wholly are described.
4.1. The virtual Rasmussen invariant
4.1.1. Formulation
Definition 4.1**.**
Given a virtual link diagram denote by the oriented smoothing of . Denote by the signed graph with a vertex for each cycle of and an edge for each classical crossing of , decorated with the sign of the crossing. The edge associated to a crossing is between the vertex or vertices associated to the cycles involved in the smoothing site of that crossing. The subgraph of formed by removing all the edges labelled with (respectively ) is denoted (respectively ). â
The graph is often called the Seifert graph of , but in order to avoid confusion with a graph defined in Section 4.2 we shall not use that term.
Definition 4.2**.**
Given a virtual knot diagram the quantities are given by
[TABLE]
The quantities and are dependent on the diagram and are not invariants of the virtual knot. â
Theorem 4.3** (Analogue of Theorem of Lobb [22]).**
For a diagram of a virtual knot
[TABLE]
Notice that the left hand side is a knot invariant whereas the right is diagram-dependent.
To prove this we require Lemma 3.1, as we have canonical generators in terms of and instead of and and the proof given in [26] relies on the sign of and .
Proof.
(of Theorem 4.3) The proof is practically identical to that of the classical case given in [22]. Form the diagram from by smoothing all the positive classical crossings of to their oriented resolution, and suppose that is the disjoint union of virtual link diagrams. Label these diagrams . Then the canonical generator splits as a tensor product of canonical generators of as
[TABLE]
Classically, can either be or where denotes the induced orientation on , as we are possibly altering the number of cycles separating others from infinity. In the virtual case, however, by construction as we use abstract link diagrams to produce the canonical generators rather than the method due to Lee.
Where the proof given in [22] invokes Theorem of [26] we invoke Lemma 3.1 as given above. â
Theorem 4.4** (Analogue of Theorem of Lobb [22]).**
If then , where is the virtual knot represented by . In fact
[TABLE]
The proof of Theorem 4.4 is identical to that of the classical case, owing to the identical behaviour of the virtual and classical Rasmussen invariants with respect to the mirror image.
4.1.2. The case
Cromwell defined homogeneous knots [8]. Here we recap his definition, which works equally well for virtual knots.
Definition 4.5**.**
A cut vertex of a graph is a vertex such that the graph obtained by removing the vertex along with its boundary edges has more connected components than . â
Definition 4.6**.**
A block of a graph is a maximal connected subgraph of containing no cut vertices. â
Definition 4.7**.**
A signed graph is homogeneous if every block of is such that all edges contained in are decorated with the same sign. â
Definition 4.8**.**
A virtual link diagram is homogeneous if is homogeneous. A virtual link is homogeneous if there exists a diagram of it which is homogeneous. â
Positive and negative virtual knots are homogeneous trivially (as possesses only one kind of decoration). In the classical case alternating knots are also homogeneous [16]. In the virtual case, however, this no longer holds. For example, the virtual knot diagram given in Figure 18(A) is alternating but not homogeneous.
Abe showed that for a classical knot diagram if and only if is homogeneous [1]. However, Abeâs proof relies on containing no loops (an edge which begins and ends at the same vertex); classically, this is always the case as the oriented resolution is the alternately coloured resolution, so that is bipartite. Virtually, however, there are knots whose oriented resolution is not the alternately coloured resolution; this is explained fully below. An example is given in Figure 19. For now, it suffices to recall that the quantity can be expressed as the first Betti number of the graph, , defined as follows.
Definition 4.9**.**
Let be associated to the virtual knot diagram . Form the graph in the following way:
- (i)
For each connected component of place a vertex, and a vertex for each connected component of . 2. (ii)
Each vertex of lies in exactly one connected component of , and exactly one connected component of . For each vertex of place an edge linking the vertices of corresponding to the connected components in which it lies. â
Proposition 4.10**.**
Let be associated to the virtual knot diagram , and be a graph obtained from by adding or removing a loop (of arbitrary sign). Further, let be the graph formed from following the method given in Definition 4.9, where and are formed in the obvious way. Then .
Proof.
It is clear that
[TABLE]
(we have only added or removed a loop) so that
[TABLE]
Further, as loops do not connect distinct vertices, two vertices are linked in if and only if they are linked in . â
In light of Proposition 4.10 it is clear that we need only consider homogeneity of up to the addition and removal of loops.
Definition 4.11**.**
Let be a signed graph and let be the graph formed by removing all loops of . Then is l-homogenous if is homogenous. A virtual knot diagram is l-homogenous if is, and a virtual knot is l-homogenous if it has an l-homogenous diagram. â
Theorem 4.12** (Analogue of Theorem of Abe [1]).**
A virtual knot diagram is l-homogeneous if and only if . Hence, for an l-homogeneous diagram of a virtual knot
[TABLE]
Proof.
Abeâs original proof yields the following statement: if is such that is loopless, then is homogenous if and only if . By Proposition 4.10 we may remove any loops from , leaving the associated unchanged. Recalling that , the first Betti number of , we obtain the desired result. â
4.2. The doubled Rasmussen invariant
4.2.1. Formulation
In formulating the bounds on the doubled Rasmussen invariant we follow much the same path as in Section 4.1. The formulae arrived at in this section exhibit important differences between those of Section 4.1, however, owing to the structural differences between MDKK homology and doubled Lee homology.
We begin by making some preliminary definitions.
Definition 4.13**.**
Let be a diagram of a virtual knot and its Gauss diagram. A classical crossing of , associated to the chord labelled in , is known as odd if the number of chord endpoints appearing between the two endpoints of is odd. Otherwise it is known as even. The odd writhe of is defined
[TABLE]
Theorem 4.14**.**
Let be a virtual knot diagram of . The odd writhe is an invariant of and we define
[TABLE]
Definition 4.15**.**
Let be a virtual knot diagram. The alternately coloured resolution of a classical crossing of is the resolution it is smoothed into in the alternately colourable smoothing of . â
Proposition 4.16** (Proposition of [27]).**
A classical crossing of a virtual knot diagram is odd if and only if itâs alternately coloured resolution is the unoriented resolution.
In the construction of MDKK homology source-sink decorations are used to ensure that the oriented resolution of a virtual knot is, in fact, alternately colourable; doubled Khovanov homology does not do so. In the definition below, therefore, we consider the graph associated to the alternately coloured smoothing of a virtual knot.
Definition 4.17**.**
Given a virtual link diagram denote by the alternately coloured smoothing of . Denote by the graph with a vertex for each cycle of and an edge for each classical crossing of , decorated with the sign and parity of the crossing: every edge is decorated with an element of , where denotes an even positive crossing, an odd positive crossing, and so on. The edge associated to a crossing is between the vertex or vertices associated to the cycles involved in the smoothing site of that crossing. The subgraph of formed by removing all the edges labelled with either or is denoted . The subgraph of formed by removing all the edges labelled with either or is denoted . â
Definition 4.18**.**
Let be a virtual knot diagram with () odd positive (negative) classical crossings. Define the quantities
[TABLE]
where denotes the number of components of a graph. â
In direct analogy to Theorem 4.3 we have the following.
Theorem 4.19**.**
Let be a diagram of a virtual knot . Then
[TABLE]
Proof.
We shall go through the proof of Theorem 4.19 in more detail than that of itâs counterpart Theorem 4.3, owing to the aforementioned differences between the theories and . The gist of the proof is unchanged, however: as computation of only requires knowledge of the partial chain complex
{CDKh_{s_{2}(K)-1}(D)^{\prime}}$${CDKh_{s_{2}(K)}(D)^{\prime}}$$d_{s_{2}(K)-1}
we ignore (by resolving them) classical crossings whose alternately coloured resolution is the [math]-resolution; such crossings are associated to outgoing maps from the alternately coloured resolution of and do not contribute to . This comes at the price, of course: we lose a large amount of the information contained in . Nevertheless, the trade is a worthwhile one, as we are able to use whatâs left to obtain bounds on .
Let be a diagram of a virtual knot , with () positive (negative) classical crossings. Further, let and , where a superscript () denotes the number of even (odd) crossings. Form a virtual link diagram, , by resolving all even positive crossings and all odd negative crossings of into their alternately coloured resolutions. (One readily observes that such crossings are those with alternately coloured resolution the [math]-resolution, as mentioned above.) We can write
[TABLE]
where is a virtual link diagram with positive and negative classical crossings (the parity of positive (negative) crossings is necessarily odd (even), of course). Further, for the alternately colourable smoothing of , we have
[TABLE]
where is the unique alternately colourable smoothing of formed by resolving all crossings into the resolution they are resolved into in .
Notice that while does not split as a tensor product of the âs, the alternately coloured generators of do. That is, if is associated to , then
[TABLE]
where is the alternately coloured generator defined by .
We have (as all negative crossings of are even), so that the highest non-trivial quantum grading of containing is , where denotes the number of cycles of . Further, as a corollary to Lemma of [27], we determine that is not of top degree, and that is the highest non-trivial degree of containing it. By Equation 4.3 and an argument directly analogous to Lobbâs [22] we obtain
[TABLE]
Recalling that , we arrive at
[TABLE]
To see that
[TABLE]
repeat the proof of Theorem 4.4, which we are free to do as the doubled Rasmussen invariant replicates the behaviour of its classical counterpart with respect to the mirror image. â
4.2.2. Simplifying
Much of the analysis used in the Section 4.1.2 may be repeated in order to characterise a case in which the formula simplifies. However, we do not recover the vanishing result as in the case of .
Definition 4.20**.**
Let be a virtual knot diagram and the graph associated to it. Recall that each edge of is decorated with exactly one element of
[TABLE]
Let and . The graph is d-homgenous if every block is decorated with elements of either or , but not both.
The diagram is d-homogenous if is d-homogenous. A virtual knot is d-homogenous if it has a d-homogenous diagram. â
Proposition 4.21**.**
Let be a virtual link diagram and the graph associated to it. Then is d-homogenous if and only if
[TABLE]
Proof.
Let denote the graph formed from in direct analogy to , as given in Definition 4.9, with and taking the place of and . The graph is bipartite as is alternately coloured. Thus it is loopless and Abeâs proof may be employed to show that is homogenous if and only if . We conclude by noticing that
[TABLE]
which follows exactly as in the case of and . â
Corollary 4.22**.**
Let be diagram of a virtual knot . If is d-homogenous then
[TABLE]
where () denotes the number of odd positive (negative) classical crossings of .
5. Computation and estimation of the slice genus
In this section we use the bounds and to compute or estimate the slice genus of a number of virtual knots. The computations are made by finding a surface of appropriate genus between the given knot and the unknot.
The following table contains the results of the analysis for the virtual knots of crossing or less in Greenâs table [13]. A blank entry denotes an unknown, and most computations of , , and (or the interval in which they lie) are made by computing , , and for the diagram given in the table. The exceptions to this are , which the author computed by hand from , and leftmost knots, for which the definition and the method of computation of are given in [27, Section ]. Further, many computations of , , and are made by spotting that the knot in question is a connect sum of two other knots, and employing the additivity of the invariants along with their invariance under flanking [27, Definition ]. (As observed in Section 2.4, and coincide for even knots, so that the invariants are buy one get one free in this case.)
Exact values of are obtained by constructing a cobordism which attains a lower bound given by , , or . Upper bounds on are obtained by constructing a cobordism of the given genus, and employing the fact that half the crossing number bounds the slice genus of a knot from above (as in the classical case) [4]. Shortly after posting a previous version of this paper to the arXiv the author learned of the work of Boden, Chrisman, and Gaudreau in which they compute or estimate the slice genus of a very large number of the virtual knots of crossings or less [4, 5]. In the table below we do not include the values of they arrive at in order to demonstrate the infomation that can be obtained using the bounds , , and the properties of the virtual and doubled Rasmussen invariants.
\csvreader
[head to column names,longtable=âcâcâcâcâcâcâcâcâ,table head=Knot l-hom. d-hom.
\endhead\endfoot
,table foot=]vknotinfo_v2.csv1=\vknot,5=\lhom,6=\dhom,7=\sinv,8=\soneinv,9=\stwoinv,10=\sgenus\vknot \lhom \dhom \sinv \soneinv \stwoinv \sgenus
From the table we are able to make some observations regarding the two extensions of the Rasmussen invariant. We see that only is able to distinguish between and . Further, there are a number of knots for which the easy to compute obstructs sliceness while the harder to compute does not. The virtual and doubled Rasmussen invariants are also able to distinguish many pairs of knots which have the same positive slice genus, showing that they are not concordant to one another.
We also give presentations of the surfaces of genus [math], , and used to determine the slice genus of the knots , , and respectively; they are contained in Figures 20, 21 and 22. Unlabeled arrows denote virtual Reidemeister moves, while those which denote -handle additions are so labelled. Red arcs between strands denote the locations of such handle additions within individual diagrams.
To conclude we list the results of similar analysis as that used to produce the previous table, this time on the virtual knots for which Boden, Chrisman, and Gaudreauâs methods are unable to obstruct sliceness but the virtual and doubled Rasmussen invariants can. The upper bounds on are those given by Boden, Chrisman, and Gaudreau [5]. As in the case of knots of or less crossings many of the computations are made by spotting connect sums.
\csvreader
[head to column names,longtable=âcâcâcâcâcâcâcâ,table head=Knot l-hom. d-hom.
\endhead\endfoot
,table foot=]vknotinfo_gaps.csv1=\vknot,5=\lhom,6=\dhom,7=\sinv,8=\soneinv,9=\stwoinv,10=\sgenus\vknot \lhom \dhom \sinv \soneinv \stwoinv \sgenus
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] T Abe. The Rasmussen invariant of a homogeneous knot. Proceedings of the American Mathematical Society , 139, 2011.
- 2[2] D Bar-Natan. Khovanovâs homology for tangles and cobordisms. Algebraic & Geometric Topology , 9, 2005.
- 3[3] D Bar-Natan and S Morrison. The Karoubi envelope and Leeâs degeneration of Khovanov homology. Algebraic & Geometric Topology , 6, 2006.
- 4[4] H U Boden, Mi Chrisman, and R Gaudreau. Virtual knot cobordism and bounding the slice genus. arxiv.org/abs/1708.05982 , 2017.
- 5[5] H U Boden, Mi Chrisman, and R Gaudreau. Virtual slice genus tables. micah 46.wixsite.com/micahknots/slicegenus. , 2017.
- 6[6] Hans U Boden and M Nagel. Concordance group of virtual knots. arxiv.org/abs/1606.06404 , 2016.
- 7[7] J S Carter, S Kamada, and M Saito. Stable Equivalence of Knots on Surfaces and Virtual Knot Cobordisms. Journal of Knot Theory and Its Ramifications , 11, 2002.
- 8[8] P R Cromwell. Homogeneous links. Journal of the London Mathematical Society , 39, 1989.
