Algorithmic simplification of knot diagrams: new moves and experiments
Abstract
This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D' with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R1, R2, R3, and another one C (together with a variant) 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.03226
 Bibcode:
 2015arXiv150803226P
 Keywords:

 Mathematics  Geometric Topology;
 57M25
 EPrint:
 38 pages, 33 figures