The Gordian complexes of knots given by 4-move.

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in 3 . 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 K 0 and for any given natural number n , there exists a family of knots { K 0 , K 1 , ... , K n } such that for any pair (K i , K j) of distinct elements of the family, the Gordian distance of knots by 4-move is d f (K i , K j) = 1. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes. [ABSTRACT FROM AUTHOR]