# The Gordian distance of handlebody-knots and Alexander biquandle colorings

## TL;DR

This paper introduces a method using Alexander biquandle colorings to establish lower bounds on the Gordian distance and unknotting number of handlebody-knots, and constructs examples with arbitrary distances.

## Contribution

It provides a novel approach to estimate handlebody-knot complexity and constructs handlebody-knots with specified Gordian distances and unknotting numbers.

## Key findings

- Lower bounds for Gordian distance and unknotting number established
- Construction of handlebody-knots with arbitrary Gordian distance and unknotting number
- Alexander biquandle colorings are effective for analyzing handlebody-knots

## Abstract

We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.