Algorithmic simplification of knot diagrams: New moves and experiments

Abstract

This paper has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram [Formula: see text] is a sequence of moves that transform [Formula: see text] into a diagram [Formula: see text] with the minimal possible number of crossings for the isotopy class of the knot represented by [Formula: see text]. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain [Formula: see text] generalizing the classical Reidemeister moves [Formula: see text], and another one [Formula: see text] (together with a variant [Formula: see text]) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.