# A multiple conjugation biquandle and handlebody-links

## TL;DR

This paper introduces a new algebraic structure called a multiple conjugation biquandle, which generalizes existing structures and provides a universal framework for defining invariants of handlebody-links.

## Contribution

The paper defines the multiple conjugation biquandle, extends n-parallel biquandle operations, and demonstrates its universality for handlebody-link invariants.

## Key findings

- Introduced the concept of multiple conjugation biquandle.
- Extended n-parallel biquandle operations to all integers.
- Showed that any biquandle induces a multiple conjugation biquandle.

## Abstract

We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of $n$-parallel biquandle operations for any integer $n$, and show that any biquandle gives a multiple conjugation biquandle with them.