The $n$-adjacency graph for knots

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.