Signature and concordance of virtual knots

TL;DR

This paper introduces Tristram-Levine signatures for virtual knots, providing new tools to study their concordance and slice genus, and extends classical invariants to all virtual knots with concrete applications and classifications.

Contribution

It develops new signature invariants for virtual knots, extends classical invariants to virtual knots, and establishes realization theorems for Seifert matrices of almost classical knots.

Findings

Signatures give bounds on the topological slice genus.

Confirmed the slice genus conjecture for most small knots.

Determined slice status for all almost classical knots with up to six crossings.

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 $\omega$-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 $\Delta(t)$ satisfying $\Delta(1)=1$ 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, $\omega$-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. Table 2 at the end summarizes our findings.