Birack Dynamical Cocycles and Homomorphism Invariants

TL;DR

This paper introduces birack dynamical cocycles, a novel algebraic enhancement of birack invariants, which improves the ability to distinguish knots and links, and connects to existing invariants.

Contribution

It defines birack dynamical cocycles as a new algebraic structure to enhance birack invariants for knots and links, providing stronger distinguishing power.

Findings

New invariant is stronger than unenhanced birack counting invariant

Examples demonstrate improved knot and link distinction

Connections established with existing knot invariants

Abstract

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack labelings of a link diagram determined by a homomorphism $p:X\to Y$ between finite labeling biracks. We provide examples to show that the new invariant is stronger than the unenhanced birack counting invariant and examine connections with other knot and link invariants.