The $n$-adjacency graph for knots

TL;DR

The paper introduces a new graph structure called $ $-adjacency graph to represent relationships between knots based on crossing changes, and proves several properties of this graph.

Contribution

It defines the $ $-adjacency graph for knots and establishes foundational results about its properties and implications.

Findings

Defined the $ $-adjacency graph to encode knot relationships.

Proved several key properties of the $ $-adjacency graph.

Provided insights into knot transformations via crossing changes.

Abstract

A knot $K$ is called $n$-adjacent to a knot $K'$ if there is a set of $n$ crossing circles $\mathcal C$ in $K$ so that a generalized crossing change at any nonempty subset of crossings in $\mathcal C$ yields $K'$. In this paper, the authors define a new graph $\Gamma_n$ to represent $n$-adjacency relationships between knots. We prove several results about this new object.