Virtual Seifert Surfaces

TL;DR

This paper introduces virtual Seifert surfaces for almost classical knots, providing a planar representation and an algorithm for their construction, which aids in computing invariants like signatures and Alexander polynomials.

Contribution

It presents the concept of virtual Seifert surfaces for AC knots and an algorithm to construct them from Gauss diagrams, facilitating invariant computations.

Findings

Virtual Seifert surfaces can be constructed from Gauss diagrams.

The method enables computation of signatures and Alexander polynomials.

The canonical genus of AC knots differs from the virtual canonical genus.

Abstract

A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $\Sigma \times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset \Sigma \times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in $\Sigma \times [0,1]$ are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing $F \subset \Sigma \times [0,1]$. 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 Stoimenow-Tchernov-Vdovina.