Geometric computations of unknotting numbers

Abstract

Unknotting number is an invariant and defined for a knot/link as the minimum number of crossing changes required to unknot a knot over all possible diagrams of knot. Unknotting numbers for torus knots and links are well known. In this thesis, we present a new approach to determine the position of unknotting number crossing changes in a toric braid B(p, q) such that the closure of the resultant braid is equivalent to the trivial knot or link. Based on this approach, we give an unknotting sequence for torus knots, an upper bound for unknotting number of some knot classes and unknotting number of more than 600 knots. One of the recent developed unknotting operations is region crossing change. Sharp upper bounds for region unknotting number of torus and 2-bridge knots are provided. Arf invariant values for some torus and 2-bridge knots are calculated. We also calculated the Trivializing number for torus knot shadows.