2017/12/15 by Chrisman, Micah · 1 citation
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1712.05715
A virtual knot that has a homologically trivial representative \mathscrK in a thickened surface Σ× [0,1] is said to be an almost classical (AC) knot. \mathscrK 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.