# Mutation invariance of the arc index for some Montesinos knots

## TL;DR

This paper investigates the mutation invariance of the arc index in certain nonalternating Montesinos knots, identifying infinite families of mutant pairs/triples sharing the same arc index, extending known invariance properties.

## Contribution

It demonstrates that infinitely many nonalternating Montesinos knots have mutant pairs/triples with identical arc indices, expanding understanding of mutation invariance beyond semi-alternating knots.

## Key findings

- Identified infinite families of mutant Montesinos knots with the same arc index.
- Extended mutation invariance properties to nonsemi-alternating, nonalternating Montesinos knots.
- Provided new examples of mutation invariance in knot theory.

## Abstract

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.