Knot Graphs and Gromov Hyperbolicity

TL;DR

This paper introduces a broad class of knot graphs, analyzes their Gromov hyperbolicity, and explores properties like homogeneity, revealing that most are not hyperbolic except for specific quotients.

Contribution

It generalizes the concept of Gordian graphs, proves non-hyperbolicity for most, and identifies conditions under which these graphs are homogeneous.

Findings

Most knot graphs are not Gromov hyperbolic.

The concordance knot graph is homogeneous.

Existence of knots with large radius balls containing no connected sum of torus knots.

Abstract

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot graphs. We also investigate the property of homogeneity, and prove that the concordance knot graph is homogeneous. Finally, we prove that that for any $n$, there exists a knot $K$ such that the ball of radius $n$ in the Gordian graph centered at $K$ contains no connected sum of torus knots.