Virtual Seifert Surfaces

Abstract

A virtual knot that has a homologically trivial representative $\mathcal{K}$ in a thickened surface $\Sigma \times [0,1]$ is said to be an almost classical (AC) knot. $\mathcal{K}$ 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.

28 pages, many figures, comments welcome!