Gromov hyperbolicity and a variation of the Gordian complex

TL;DR

This paper introduces new knot-based simplicial complexes using invariants and local moves, demonstrating that a specific complex related to the Alexander-Conway polynomial and Delta-move is Gromov hyperbolic and quasi-isometric to the real line.

Contribution

It presents a novel class of simplicial complexes for knots, extending the Gordian complex, and proves hyperbolicity and quasi-isometry properties for a particular case.

Findings

The complex is Gromov hyperbolic.

The complex is quasi-isometric to the real line.

New invariants and moves lead to generalized complexes.

Abstract

We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line.