The $n$-adjacency graph for knots

A knot $K$ is called $n$-adjacent to a knot $K^{\mathrm{\prime }}$ 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 $K^{\mathrm{\prime }}$. In this paper, the authors define a new graph ${\mathrm{\Gamma }}_n$ to represent $n$-adjacency relationships between knots. We prove several results about this new object.