TAILIEUCHUNG - Computation of gordian distances and H2-gordian distances of knots

In this paper we discuss the possibility of computing unknotting number from minimal knot diagrams, Bernhard-Jablan Conjecture, unknown knot distances between non-rational knots and of searching minimal distances by using a graph with weighted edges representing knot distances. Since topoizomerazes are enzymes involved in changing crossing of DNA, knot distances can be used to study topoizomerazes actions. | Yugoslav Journal of Operations Research 25 (2015), Number 1, 133-152 DOI: COMPUTATION OF GORDIAN DISTANCES AND H2-GORDIAN DISTANCES OF KNOTS Ana ZEKOVIĆ Faculty of Mathematics, University of Belgrade, Serbia azekovic@ Received: November 2013 / Accepted: December 2013 Abstract: In this paper we discuss the possibility of computing unknotting number from minimal knot diagrams, Bernhard-Jablan Conjecture, unknown knot distances between non-rational knots and of searching minimal distances by using a graph with weighted edges representing knot distances. Since topoizomerazes are enzymes involved in changing crossing of DNA, knot distances can be used to study topoizomerazes actions. We compute some undecided knot distances 1 known from the literature, 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 the program LinKnot, based on Conway notation and nonminimal representations of knots. Keywords: Unknotting number, Gordian distances, Smoothing distances, Weighted graph search. MSC: 57M25, 57M27. 1. INTRODUCTION The question of unknotting numbers, or Gordian numbers is one of the most difficult in knot theory [1, 2]. In order to compute unknotting numbers, we need a link surgery. In every crossing of a knot, it is possible to make a crossing change (Fig. 1a): to transform an overcrossing to undercrossing or vice versa. Crossing change is unknotting operation. Definition 1. The unknotting number u(D) of a knot diagram D is the minimal number of crossing changes on the diagram required to obtain a diagram representing an unknot; 134 A. Zeković / Computation Of Gordian Distances The uM(K) of a knot K in R3 is the minimum of u(D) over all minimal crossing number diagrams D representing knot K; The unknotting number u(K) of a knot K in R3 is the minimum of u(D) over all diagrams D .

• Toán học
• Unknotting number
• Gordian distances
• Smoothing distances
• Weighted graph search
• Minimal knot diagrams