Quandle colorings vs. biquandle colorings

TL;DR

This paper establishes a direct correspondence between biquandle and quandle colorings for classical and surface links, showing their invariants are essentially equivalent, and interprets biquandle cocycle invariants via shadow quandle cocycles.

Contribution

It provides an explicit bijection between biquandle and quandle colorings and demonstrates the equivalence of their homotopy invariants for classical and surface links.

Findings

Biquandle and quandle colorings are in one-to-one correspondence for classical/surface links.

Biquandle homotopy invariants are equivalent to quandle homotopy invariants.

Biquandle cocycle invariants can be interpreted as shadow quandle cocycle invariants.

Abstract

Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for classical/surface links so far. In this paper, we give an explicit one-to-one correspondence between biquandle colorings and quandle colorings for classical/surface links. We also show that biquandle homotopy invariants and quandle homotopy invariants are equivalent. As a byproduct, we can interpret biquandle cocycle invariants in terms of shadow quandle cocycle invariants.