Behavior of Gordian graphs at infinity

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.