# Shadow biquandles and local biquandles

## TL;DR

This paper explores the relationship between shadow biquandles and local biquandles, showing their (co)homology groups and cocycle invariants are equivalent, and connects these structures to knot theory and existing cocycle theories.

## Contribution

It establishes an isomorphism between shadow biquandle and local biquandle (co)homology groups and relates cocycle invariants to Niebrzydowski's theory and Mochizuki's cocycles.

## Key findings

- (Co)homology groups of shadow biquandles are isomorphic to those of local biquandles.
- Cocycle invariants for links and surface-links coincide between shadow and local biquandles.
- Certain cocycles can be induced from Mochizuki's cocycles.

## Abstract

Given a shadow biquandle $(B,X)$ composed of a biquandle $B$ and a strongly connected $B$-set $X$, we have a local biquandle structure on $X$. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. These results imply that for some cases, the Niebrzydowski's theory in [14, 15, 16] for knot-theoretic ternary quasigroups is the same as shadow biquandle theory. We also show that some local biquandle $2$- or $3$-cocycles and some $1$- or $2$-cocycles of the Niebrzydowski's (co)homology theory can be induced from Mochizuki's cocycles.