Virtual Seifert surfaces.

A virtual knot that has a homologically trivial representative 𝒦 in a thickened surface Σ × [ 0 , 1 ] is said to be an almost classical (AC) knot. 𝒦 then bounds a Seifert surface F ⊂ Σ × [ 0 , 1 ]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in Σ × [ 0 , 1 ] are difficult to construct. Here, we introduce virtual Seifert surfaces of AC knots. These are planar figures representing F ⊂ Σ × [ 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. [ABSTRACT FROM AUTHOR]