The Gordian complexes of knots given by 4-move

Abstract

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in [Formula: see text]. Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot [Formula: see text] and for any given natural number [Formula: see text], there exists a family of knots [Formula: see text] such that for any pair [Formula: see text] of distinct elements of the family, the Gordian distance of knots by 4-move is [Formula: see text]. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes.