Link invariants from finite biracks

TL;DR

This paper extends the counting invariant for finite racks to finite biracks, introducing a new family called $(\tau,\sigma,\rho)$-biracks and exploring enhancements using various algebraic and combinatorial tools.

Contribution

It generalizes the counting invariant to finite biracks and introduces a new family of biracks called $(\tau,\sigma,\rho)$-biracks, expanding the algebraic framework for link invariants.

Findings

Extended counting invariant to finite biracks

Introduced $(\tau,\sigma,\rho)$-biracks as a generalization

Developed enhancements using writhe vectors, image subbiracks, and birack polynomials

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.