Behavior of Gordian graphs at infinity

TL;DR

This paper investigates the large-scale structure of Gordian graphs in knot theory, specifically their behavior at infinity for various local moves, providing a comprehensive description of their ends and unbounded components.

Contribution

It offers a complete characterization of the behavior at infinity for Gordian graphs under rational, C(n)-, and H(n)-moves, extending understanding of their global properties.

Findings

Describes the ends of Gordian graphs for rational, C(n)-, and H(n)-moves.

Characterizes the behavior at infinity for all local moves with the infinite neighborhood of the unknot.

Provides a unified perspective on the large-scale structure of Gordian graphs in knot theory.

Abstract

The present paper refers to the knot theory and is devoted to the study of global properties of Gordian graphs of various local moves. In 2005, Gambaudo and Ghys raised the question of the behavior at infinity of the crossing change Gordian graph. They proposed studying its "ends", that is, unbounded connected components of complements of bounded subsets. We provide a complete description of the behavior at infinity for local moves from three well-known infinite families, namely, rational moves, $C(n)$-moves, and $H(n)$-moves (note that each of the first two families contains the crossing change). Also, in 2005, March\'e gave a different perspective on the behavior of Gordian graphs at infinity, proposing to consider complements of finite subsets. We describe the behavior at infinity in this sense for all local moves with the infinite neighborhood of the unknot in the corresponding Gordian graph.