The Gordian complex of theta-curves

Abstract

In this paper, we study the Gordian metric on the set of all theta-curves and give a lower bound of it. We define the Gordian complex of theta-curves, which is a simplicial complex whose vertices consist of all theta-curves in the 3-dimensional Euclidean space [Formula: see text]. We show that for any given theta-curve [Formula: see text], there exists an infinite family of theta-curves containing [Formula: see text] such that the Gordian distance between any two distinct members of this family is equal to one.