Invariants of virtual knots and spatial graphs

Abstract

Virtual knots (links) and spatial graphs are two natural extensions of knots. In this thesis, we discuss about invariants of virtual knots (links) and spatial graphs. Using crossing change operation and virtualization, we define a new virtual link invariant called unknotting index, whose idea is an extension of usual unknotting number of knots. Using the writhe invariant and span invariant, we provide a lower bound on the unknotting index. With the help of this lower bound, the unknotting index of some families of virtual knots/links is determined. Also we prove that for any two positive integers n and m, there exists an infinite family of virtual knots with unknotting index (n;m). Further, we introduce two sequences of two-variable polynomials fLn K(t; `)g1 n=1 and fFnK (t; `)g1 n=1, expressed in terms of index value of a crossing and n-th dwrithe value of a virtual knot K, where t and ` are variables. Based on the fact that n-th dwrithe is a flat virtual knot invariant, we prove that Ln K and FnK are virtual knot invariants, and Kauffman’s affine index polynomial is a particular case of these invariants. Using Ln K we provide conditions to specify when a virtual knot does not admit cosmetic crossing change. For spatial graphs, we introduce Gauss diagrams and discuss their equivalence using generalized Reidemeister moves. In [40], A. Kawauchi discuss about unknotting number and 􀀀-unknotting number for a spatial graph. We define based unknotting number and based 􀀀-unknotting number for a spatial graph, and discuss relation between these unknotting numbers. In [19], R. Hanaki introduced the notion of pseudo diagram and the trivializing number of spatial graphs whose underlying graphs are planar. We generalize the concept of trivializing number without considering the assumption that the underlying graphs are planar, and define 􀀀-trivializing number and based 􀀀-trivializing number. Finally, we discuss relations among 􀀀-unknotting numbers and 􀀀-trivializing numbers.