New unknotting invariants and geometric complexes of virtual knots

Abstract

In this thesis, we formulated two new local moves on virtual knot diagrams that we name arc shift and region arc shift, respectively. This whole thesis deals with studying these two moves, arc shift and region arc shift moves. We show that both arc shift and region arc shift are unknotting operations for virtual knots. Based on these results, unknotting invariants, arc shift number, and region arc shift number are defined, respectively, and denoted by A(K) and R(K). Further, we establish that R(K) F(K), where F(K) denotes the forbidden number of virtual knot K. We provide a lower bound on arc shift number A(K) in terms of odd writhe J(K). Using the lower bound, we compute A(K) for a few infinite families of virtual knots whose virtual bridge number is one. For n 1, we show the existence of infinitely many virtual knots with arc shift number A(K) = n. Further, we study the variation of n-writhes Jn(K) under arc shift move. We also show that Jn(K) varies randomly in the sense that it may increase or decrease by a random integer while applying a single arc shift move. As a consequence of the variation of Jn(K), we show that coefficients and degree of affine index polynomial vary unboundedly under arc shift move. Further, we formulate and study the Gordian complexes of knots and virtual knots defined by region crossing change and arc shift move, respectively. We study the structure of simplexes and show the existence of arbitrarily high dimensional simplexes in both the Gordian complexes defined. In the Gordian complex, by region crossing change, we answer in affirmative the question about the existence of an n-simplex containing a given m-simplex for each n > m, when m 2.