Concordance group of virtual knots

Abstract

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and ω-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots.The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial ∆(t) satisfying ∆(1) = 1 occurs as the Alexander polynomial of an almost classical knot.In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, ω-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four.

Virtual Seifert surfaces

“…Next, step (3) is obtained from step (2) by three moves of type (C) and some re-positioning of the bands. Performing move (A) together with move (B) gives the virtual Seifert surface seen in step (4). The dotted curve in step (5) encloses a region over which the projector χ is one-to-one.…”

“…The dotted curve in step (5) encloses a region over which the projector χ is one-to-one. Thus, we may perform moves from Figure 18 (see bottom left and right) on the virtual Seifert surface in step (4). Lastly, step (6) is obtained from step (5) via isotopies of F and moves of type (D).…”

“…Here we consider two related questions: (1) how does one draw a Seifert surface for a homologically trivial knot in a thickened surface Σ × [0, 1], where Σ 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 [1] 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.…”

“…We apply this to computing slice obstructions. Other applications of virtual Seifert surfaces will appear elsewhere (see [1] v2).…”

“…Section 5 shows how to manipulate virtual Seifert surfaces into virtual band presentations. Section 6 applies virtual Seifert surfaces to computing the slice obstructions from [1]. In Section 7, the virtual canonical 3-genus is studied.…”