Virtual Knot Invariants from Group Biquandles and Their Cocycles

A group-theoretical method, via Wada’s representations, is presented to distinguish Kishino’s virtual knot from the unknot. Biquandles are constructed for any group using Wada’s braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings. The purposes of this paper include defining biquandle structures on groups, and giving a grouptheoretic proof that Kishino’s virtual knot is non-trivial. A biquandle structure or a birack structure is related to solutions to the set-theoretic Yang-Baxter equation (SYBE). Given an invertible solution that satisfies an additional condition (corresponding to a Reidemeister type I move), we obtain a biquandle, and every biquandle gives a solution to the SYBE. Most examples that were known up to this point came from generalizations of the Burau representation. The principal examples that we consider in this paper come from Wada’s representations of braid groups as free group automorphisms. Our first example indicates that one of these representations can be used to distinguish Kishino’s virtual knot from the unknot. By abelianizing such groups, we recover an analog of Burau matrix. Using these examples, we construct and calculate cocycle invariants that come from the homology theory of biquandles. As applications we give obstructions to checkerboard colorability and (mod 2)-Alexander numberings of arcs of virtual knots. In Section 2 we examine Wada’s group invariants for virtual knots. The biquandle structures are defined on any group in Section 3 using Wada’s representations, and colorings of virtual knot diagrams by such biquandles are studied. Cocycle invariants are constructed and applied in Section 4.