# Linear extensions of multiple conjugation quandles and MCQ Alexander pairs

## TL;DR

This paper introduces MCQ Alexander pairs to describe linear extensions of multiple conjugation quandles, connecting algebraic structures from knot and handlebody-knot theory.

## Contribution

It extends the concept of Alexander pairs to multiple conjugation quandles, providing a new algebraic framework for handlebody-knot theory.

## Key findings

- Defined MCQ Alexander pairs for multiple conjugation quandles
- Established a correspondence between linear extensions and MCQ Alexander pairs
- Enhanced algebraic tools for handlebody-knot theory

## Abstract

A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, where a multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory.