A Combinatorial approach to Knot theory and its applications

Abstract

In this thesis we will be looking at knots and links using the combinatorial structure of various invariants. The focus will be the Jones polynomial, Fraction invariant and the Alexander Polynomial. While we develop combinatorial definitions of these invariants, the aim is not just to classify the knots but to understand various other aspects of these knots and links which could be derived from these combinatorial structures. Further, we also look at a few applications of these combinatorial structures in Biology, specifically in understanding linear polymer chains and related phenomenon.