# Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

## TL;DR

This paper explores the structure of Gordian complexes for knots and virtual knots using region crossing changes and arc shift moves, demonstrating high-dimensional simplices and constructing infinite families with specific move distances.

## Contribution

It introduces Gordian complexes based on region crossing change and arc shift move, showing their high-dimensional simplices and constructing infinite families with controlled move distances.

## Key findings

- Existence of arbitrarily high dimensional simplices in both complexes.
- Construction of infinite families with pairwise distance one.
- Virtual knots share the same affine index polynomial.

## Abstract

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that that the constructed virtual knots have the same affine index polynomial.