Quotients of the Gordian and H(2)-Gordian graphs

TL;DR

This paper studies quotients of Gordian and H(2)-Gordian graphs of knots under various invariants, showing they are Gromov hyperbolic and characterizing their isomorphism types, including a complete graph structure for certain link quotients.

Contribution

It introduces new quotient graphs of knot Gordian graphs based on invariants and proves their hyperbolicity and specific isomorphism properties, including a complete graph structure.

Findings

Quotient graphs are Gromov hyperbolic.

H(2)-Gordian graph of links modulo span of Jones polynomial is a complete graph.

Characterization of isomorphism types of quotient graphs.

Abstract

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.