New invariants for virtual knots via spanning surfaces

TL;DR

This paper introduces new invariants for virtual knots using spanning surfaces, including Alexander polynomials and Floer homology, which help distinguish virtual knots from classical ones and analyze their properties.

Contribution

It defines novel spanning surface-based invariants and extends Floer homology to virtual knots, providing tools to distinguish and obstruct certain virtual knot properties.

Findings

New Alexander polynomial extends to virtual knots and detects non-classicality.

Constructed Floer homology for knots in thickened surfaces, with limitations on stabilization invariance.

Used the δ-invariant to obstruct knots from being stabilizations of others.

Abstract

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens-Golla $\delta$-invariant to obstruct a knot from being a stabilization of another.