Link invariants from finite biracks

Abstract

A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(\tau,\sigma,\rho)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.