Bikei invariants and gauss diagrams for virtual knotted surfaces

Abstract

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in [Formula: see text]; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.