Topological Invariants of Generalized Weaving Knots and Theta-Curves

Abstract

The main objective of this thesis is to study determinants and unknotting numbers for certain families of weaving knots and their generalizations. Besides that, we also study the Gordian complex of theta-curves. The first part of this thesis presents determinant formulae for the 3-strand weaving knots, weaving knots of repetition index two, twisted generalized hybrid weaving knots, and 5-strand spiral knots. Further, we calculate the dimension of the first homology group with coefficients in Z3 of the double branched cover of the 3-sphere S3 over 3-strand weaving knots and weaving knots of repetition index two, respectively. As a consequence, we obtain a lower bound of the unknotting number for 3-strand weaving knots in certain cases. Some upper bounds of the unknotting numbers of 3-strand weaving knots and weaving knots of repetition index two are also discussed. In the second part of this thesis, we extend the notion of the Gordian metric to the set of theta-curves and give a lower bound of the same. Then we define the Gordian complex of theta-curves and study its structural properties. More precisely, the existence of an n-dimensional simplex of theta-curves for any n is shown. We also prove that given any theta-curve, there exists an infinite family of theta-curves containing the given theta-curve such that the Gordian distance between any pair of distinct members of this family is one.