Virtual Seifert Surfaces
Abstract.
A virtual knot that has a homologically trivial representative in a thickened surface is said to be an almost classical (AC) knot. then bounds a Seifert surface . Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing . An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of StoimenowTchernovVdovina.
Key words and phrases:
Virtual Seifert surfaces, signature functions, canonical Seifert genus.2010 Mathematics Subject Classification:
Primary: 57M25, Secondary: 57M27For an oriented knot or link in , H. Seifert gave in [seifert] a simple algorithm for constructing a compact oriented surface bounded by , now called a Seifert surface of . As is well known, Seifert surfaces appear prominently in the computation of many link invariants, such as Alexander polynomials, signature functions, and Milnor invariants. Seifert surfaces can also be used to study homologically trivial knots in other manifolds (e.g. see [ct, UK]). The construction of a Seifert surface, however, can be both mathematically and artistically challenging in the more general setting.
Here we consider two related questions: (1) how does one draw a Seifert surface for a homologically trivial knot in a thickened surface , where is compact, connected, and oriented, and (2) how can such surfaces be employed in the study of virtual knots? These issues arise, for example, in the computation of the directed signature functions [bcg2] of Boden, Gaudreau, and the author. Directed signatures give bounds on the slice genus for those virtual knots that can be represented by a homologically trivial knot in some thickened surface. To compute them, a choice of Seifert surface must be made. Hence, a systematic means for producing Seifert surfaces in is needed in order to extract the geometric content of directed signature functions.

A virtual knot that admits a homologically trivial representative in some is called an almost classical (AC) knot. Here we define virtual Seifert surfaces for AC knots. A virtual Seifert surface is a planar figure that depicts a Seifert surface , similar to the manner in which a virtual knot diagram is a planar figure depicting a knot . An example of a virtual Seifert surface is shown above. Our main result is an algorithm for constructing virtual Seifert surfaces from the Gauss diagram of an AC knot. We apply this to computing slice obstructions. Other applications of virtual Seifert surfaces will appear elsewhere (see [bcg2] v2).
Recall the Seifert surface algorithm for a classical knot. Starting with a knot diagram in , perform the oriented smoothing at each crossing. This results in disjoint closed curves called Seifert cycles. Each curve bounds a disc in , which forms a subsurface of the Seifert surface. The discs are placed at different heights in and halftwisted bands are attached to the discs at the crossings of . The result is a Seifert surface of the knot .
The theory of virtual Seifert surfaces given here proceeds along similar lines. For every Gauss diagram , there is a naturally associated surface to , called the Carter surface. The Gauss diagram can be drawn as a knot diagram on . For an AC knot, is homologically trivial on . The homology chain complex of and the homology class of each Seifert cycle of can be computed directly from . Since , linear combinations of the Seifert cycles bound subsurfaces of . The subsurfaces are themselves linear combinations of handles in the standard handle decomposition of the Carter surface. The linear combinations can be explicitly drawn in the plane, although pieces of the surface may need to pass over one another virtually. The virtual Seifert surface is again completed by attaching halftwisted bands at the crossings.
A useful feature of classical Seifert surfaces is that they can be deformed into discband presentations via pictures in the plane (see e.g. [on_knots]). The advantage of this lies in the fact that the linking numbers in a Seifert matrix are easier to compute when a surface takes this form. Here we will show how to manipulate virtual Seifert surfaces into virtual band presentations. The deformations can also be accomplished through planar pictures. The entries of the Seifert matrix are in this case virtual linking numbers and the calculation of invariants then proceeds as usual.
The genus of a classical knot is the minimal genus among all Seifert surfaces that it bounds. A Seifert surface constructed from a knot diagram with Seifert’s algorithm is said to be canonical. The canonical 3genus is the minimal genus among all canonical Seifert surfaces of a knot. Moriah [moriah] proved that the difference between the canonical genus and genus of a classical knot can be arbitrarily large. A natural question to ask is whether the virtual Seifert surface algorithm applied to some virtual knot diagram of a classical knot can produce a surface of genus smaller than its canonical genus. Here we show that this is impossible: the smallest genus among all virtual Seifert surfaces of a classical knot is the classical canonical genus. This is accomplished by building on important work of Bodenetal.[acpaper], Kauffman [lou_cob], StoimenowTchernovVdovina [sto_canon], and Tchernov [chernov_proj]. As a clarification of terminology, the smallest genus among all virtual Seifert surfaces of an AC knot, which we will call the virtual canonical 3genus, is a fundamentally different concept from the canonical genus defined in [sto_canon]. In fact, we will see an example where they are unequal. For classical knots, we prove the two notions coincide with the classical canonical 3genus.
The organization of this paper is as follows. Section 1 reviews virtual and almost classical knots. Section 2 discusses a method of computing the homology chain complex of an AC knot. Section 3 gives precise definitions of virtual band presentations and virtual Seifert surfaces. The virtual Seifert surface algorithm is given in Section LABEL:sec_vss. Section LABEL:sec_step_6 shows how to manipulate virtual Seifert surfaces into virtual band presentations. Section LABEL:sec_comp applies virtual Seifert surfaces to computing the slice obstructions from [bcg2]. In Section LABEL:sec_canon, the virtual canonical genus is studied. In this paper, decimal numbers, such as 5.2025, refer to the virtual knot names from Green’s tabulation [green]. The ones digit denotes the classical crossing number. Three examples are used throughout to illustrate the virtual Seifert surface algorithm: 4.99, 5.2025, and 6.87548. Henceforth, we set .
1. Background
1.1. Virtual knots
Virtual knots were introduced by L. H. Kauffman in the mid 1990s [KaV]. They have several equivalent definitions. A virtual knot diagram is a generic immersion , where the double points are decorated as classical crossings or virtual crossings. A virtual crossing is denoted by a small circle around the double point. Two virtual knots and are said to be equivalent, denoted , if one may be obtained from the other by a finite sequence of extended Reidemeister moves (see Figure 1). An equivalence class of diagrams is called a virtual knot.

The Gauss diagram of a virtual knot diagram is found by connecting the preimages of the double points of the classical crossings of by an arrow. The arrow is directed from the overcrossing arc to the undercrossing arc (see Figure 2). Each arrow of is also marked with the sign of the local writhe of the crossing. Any two virtual knot diagrams that have the same Gauss diagram are equivalent [GPV].
A virtual knot diagram can then be viewed as the result of an attempt to draw a configuration of classical crossings in a Gauss diagram as a knot diagram in . The inability to do so for a particular configuration necessitates the addition of crossings not specified by the Gauss diagram and hence are marked as virtual. Every Gauss diagram, however, can be drawn as a knot diagram on a higher genus surface. One way to see this is to use the Carter surface of a virtual knot diagram [CKS].
The Carter surface is constructed from a virtual knot diagram using a handle decomposition. A handle (i.e. a disc) is centered at each classical crossing. The handles are untwisted bands attached to the handles along the arcs between the classical crossings of . Thus the arcs of are the cores of the 1handles. At a virtual crossing, the handles pass over and under one another, as in Figure 3. The resulting surface is closed by attaching handles along each boundary component (see Figure 4). Observe that the handle decomposition is determined only by the Gauss diagram of . Furthermore, changing any of classical crossings of from over to under (or vice versa) gives a Carter surface with an identical handle decomposition. This fact will be used later.







A Gauss diagram can also be constructed from a knot diagram on any closed oriented surface. If a handle is added to this surface so that it is disjoint from the knot diagram, then the Gauss diagram of the knot remains unchanged. The operation of adding a handle is called stabilization. The inverse operation is called destabilization. The Gauss diagram of a knot diagram on a surface is also unchanged by orientation preserving diffeomorphisms of surfaces. Thus, virtual knots can be interpreted as knots in thickened surfaces, considered equivalent up to ambient isotopy, orientation preserving diffeomorphisms of surfaces, and stabilization/destabilization.
The first step in the classical Seifert surface algorithm is to perform the oriented smoothing at every crossing. For future use in constructing virtual Seifert surfaces, we now describe the effect that different kinds of smoothing have on a Gauss diagram. The oriented smoothing (see Figure 5, far left and far right) at one crossing splits the knot diagram into a twocomponent oriented link. The disoriented smoothing (see Figure 5, middle two pictures) at one crossing yields an unoriented knot diagram. To smooth a Gauss diagram, first delete a small neighborhood of each arrow endpoint and then reconnect the ends. The manner of reconnecting for each type of smoothing is also shown in Figure 5. Lastly, recall that a smoothing at a crossing may be either an smoothing or a smoothing. In an oriented knot diagram, an or smoothing may be either oriented or disoriented, depending on the crossing sign. The resulting four possibilities are labeled in Figure 5.




1.2. Almost Classical Knots
For an oriented knot in a manifold , a Seifert surface of is a compact connected oriented surface such that , where the orientation of induces the given orientation of . Suppose is a virtual knot, is a representative of in some thickened surface , and that bounds a Seifert surface . Then is said to be almost classical (AC). Almost classical knots were originally defined by SilverWilliams [silwill] as those virtual knots admitting a diagram with an Alexander numbering. Boden et al.[acpaper] tabulated the distinct AC knots having classical crossing number at most . There are such nontrivial AC knots in total; their Gauss diagrams can be found in Figure 20 at the end of [acpaper].
There are four equivalent definitions of AC knots that are each useful under different circumstances. For a proof of said equivalence, see Sections 5 and 6 of [acpaper]. The conditions are:

admits a diagram with an Alexander numbering.

has a homologically trivial representative in some thickened surface (and in particular, admits a diagram which is homologically trivial on its Carter surface)

has a representative that bounds a Seifert surface in some thickened surface .

admits a Gauss diagram such that every arrow has index .
Condition (4) provides the simplest method to prove that a given virtual knot is AC. Geometrically, the index of a crossing of a knot diagram on a surface is (up to sign) the algebraic intersection number of the two closed curves obtained by performing the oriented smoothing at . This can be computed combinatorially from a Gauss diagram (see e.g. [acpaper, henrich]).
By an AC diagram (respectively, AC Gauss diagram) of a virtual knot, we mean a diagram (respectively, Gauss diagram) that satisfies one of the equivalent conditions for a virtual knot to be AC.
A constructive proof that was given in [acpaper]. Here we sketch the argument for later use. Suppose a diagram of a homologically trivial knot on the Carter surface is given. First perform the oriented smoothing at each crossing of . This creates a family of disjoint simple closed curves on . These are called the Seifert cycles of . Since is homologically trivial on , so is . Then there is a collection of connected compact oriented subsurfaces of such that for all and (see [acpaper], Proposition 6.2). A Seifert surface is constructed by placing the surfaces at different heights in as necessary and then attaching halftwisted bands at the smoothed crossings of .
If , this gives the Seifert surface algorithm for classical knots. In this case, each may be chosen to be a disc. This is not true for AC knots in general. Indeed, the genus of each might be any nonnegative integer and the number of boundary components of might be any natural number. This leads to an ambiguity in the constructive existence proof for Seifert surfaces from [acpaper]: as there are many options, how does one find linear combinations of the Seifert cycles that bound subsurfaces of ? The virtual Seifert surface algorithm builds off the constructive proof from [acpaper] and reduces this ambiguity to a homological calculation.
2. The homology of the Carter surface of AC knots
Here we give a method for computing the homology chain complex of the Carter surface for an AC knot diagram on . This will be employed in the virtual Seifert surface algorithm. Let be the free abelian group generated by the handles of the Carter surface (see Section 1) and the th boundary map. The 0handles of correspond to the classical crossings of and hence correspond to the arrows of . The cores of the handles are the arcs between the classical crossings, and hence correspond to the arcs of between the arrow endpoints. Denote the handles of by . Our main interest is . To compute , it is necessary to write as a linear combination of the . First we review some terminology.
A virtual knot diagram is said to be alternating if the classical crossings alternate successively between over and under while traversing the diagram (note that virtual crossings are ignored). Also recall that the all state (all state) is the set of cycles obtained by performing the smoothing (respectively, smoothing) at every classical crossing. The lemma below relates the computation of for an AC Gauss diagram to the all and all states of any alternating diagram that is obtained from by crossing changes. Recall from Section 1.1 that and then have identical Carter surfaces.
Lemma 1.
Every AC Gauss diagram can be transformed to an alternating diagram by a finite number of crossing changes. For any such there is a onetoone correspondence between the cycles and the components of the all state and the all state.
Proof.
An AC knot is homologically trivial on its Carter surface. All such virtual knots are checkerboard colorable. N. Kamada proved in [nkam], Lemma 7, that every checkerboard colorable virtual knot can be transformed to a alternating diagram by a finite number of crossing changes. We remark that is then found by changing both the direction and sign of the corresponding arrows of . Note that any crossing change on an AC knot diagram does not affect the handle decomposition of the Carter surface. Finally, it follows from Figure 6 that the boundary of each handle is either a component in the all state of or the all state of . ∎
We may now write as a matrix in the bases of and of . After making into an alternating diagram , find the components of the all and all states. Figure 5 indicates the effect of the smoothing on a signed arrow. The components are . To determine the signs of the contributions of the to , we must orient each . Draw a classical crossing of and label it with all of the  and handles that are incident to it. Give each handle the counterclockwise orientation. Each edge of is then either , according to whether it “goes with” () or “goes against” (). Transferring the orientation of the edge back to indicates the orientation of each edge of . This orients all the handles incident the chosen crossing in both the all and all states. To orient all the other handles, observe that in each state, the orientations must alternate while traversing the circle of .
Example (4.99): A Gauss diagram of is given in Figure 7. This may be made alternating by changing the direction and sign of the two signed arrows. The resulting diagram has all signed arrows. The all state is then found by choosing the disoriented smoothing at each crossing (see Figure 5). The all state takes, in this case, the oriented smoothing at each crossing. To compute , label the eight arcs of as . These handles generate . Similarly, label each of the components of the all and all states: . Orient the handles by drawing one crossing and all the handles incident to it (see Figure 7, top right). The columns of can then be easily read off the figure. We use the symbol to denote the transpose.
Two additional examples of the computation of are given in Figures LABEL:fig_5pt2025 and LABEL:fig_6pt87548.





3. Virtual Band Presentations and Seifert Surfaces
To compute a Seifert matrix for a classical knot in , one begins with a Seifert surface of the knot and then performs an isotopy to a discband presentation. Seifert matrices, and hence invariants such as the Alexander polynomial and the signature, can then be computed from the discband presentation. In this section, we give the formal definitions of virtual band presentations and virtual Seifert surfaces. It will be shown that every AC knot has a virtual Seifert surface and every virtual Seifert surface can be deformed to a virtual band presentation.
3.1. Virtual Band Presentations
Let be an oriented disc embedded in . In , attach smooth arcs to so that the set of points are all distinct. Furthermore, we assume that the arcs intersect each other and themselves only in virtual or classical crossings. This can be viewed as a virtual tangle drawn in the disc . Fatten each arc slightly into a immersed band in so that there is a classical crossing of bands at each classical crossing and a virtual crossing of bands at each virtual crossing (see Figure 8). The union of the disc together with these bands is called a virtual band presentation with underlying virtual tangle . The orientation of determines the orientation of the immersed surface .
Note that constitutes a virtual link diagram , where the natural choices of over, under, and virtual crossings of arcs are made at each double point. We will say virtually bounds . An example of a virtual band presentation is given in Figure 9. It will be seen later that its virtual boundary is .
Virtual band presentations were introduced in [bcg2] (see v2). They were used to show that if is a pair of integral matrices satisfying and , then there is an AC knot having as its pair of directed Seifert matrices (for definitions, see Section LABEL:sec_link ahead). The AC knot that realizes the pair virtually bounds a virtual band presentation. Conversely, our goal is to draw a Seifert surface from an AC Gauss diagram.
3.2. Virtual Seifert Surfaces
A useful feature of Seifert surfaces for classical knots is that they can be deformed into discband presentations via pictures in the plane. In order to do this in the virtual setting, it is necessary to have a planar representation of the ambient space . Any compact oriented surface with at least one boundary component can be immersed into . The image of the immersion can be thought of intuitively as a screen onto which an actor (e.g. a knot or Seifert surface in ) can be projected. Places where the immersed surface overlaps itself are the only places where the images of or can have virtual crossings. A precise definition of projector and screen is needed to define virtual Seifert surfaces (Definition LABEL:defn_vss) and manipulate them (Section LABEL:sec_step_6).
Let be a compact connected oriented smooth surface with . Fix a handle decomposition of into handles and handles , where each handle is attached to . Suppose that is an orientation preserving immersion that embeds the disjointly in the plane and immerses the in such that images of the handles can intersect themselves and each other only in virtual crossings of bands (see Figure 8, left). Then a virtual screen (or simply screen) is the image of . The map is a projector onto . The overlaps of the immersed handles are called the virtual regions. Each virtual region is the image of two embedded discs and for some . In Figure LABEL:fig_virt_screen_defn, and is the restriction of the canonical projection onto the first factor, i.e. .