Computation of gordian distances and H2-Gordian distances of knots

Abstract

One of the most complicated problems in Knot theory is to compute unknotting number. Hass, Lagarias and Pippenger proved that the unknotting problem is NP hard. In this paper we discuss the question of computing unknotting number from minimal knot diagrams, Bernhard-Jablan Conjecture, unknown knot distances between non-rational knots, and searching for minimal distances by using a graph with weighted edges, which represents knot distances. Since topoizomerazes are enzymes involved in changing crossing of DNA, knot distances can be used to study topoizomerazes actions. In the existing tables of knot smoothing, knots with smoothing number 1 are computed by Abe and Kanenobu [27] for knots with at most n = 9 crossings, and smoothing knot distances are computed by Kanenobu [26] for knots with at most n = 7 crossings. We compute some undecided knot distances 1 from these papers, and extend the computations by computing knots with smoothing number one with at most n = 11 crossings and smoothing knot distances of knots with at most n = 9 crossings. All computations are done in LinKnot, based on Conway notation and non-minimal representations of knots.