Study of Some Problems in Knot Theory

Abstract

The main purpose of this thesis is to obtain some new cri- teria of periodicity of links depending on some polynomial link invariants. Speci cally, we try to have some progress toward solving a well-known conjecture in knot theory that the period of a periodic alternating link divides its crossing number. In Chapter 1: We give a historical introduction of knot theory along the time together with an overview of the main concepts and de nitions. Also, we recall some of classical link invariants and the Reidemeister moves that we use many times in our work. In Chapter 2: We study some polynomial invariants in one variable as the Kau man bracket and the Jones polynomial. We study di erent approaches of the Jones polynomial. We evaluate the Jones polynomial of some links by hand. We end this chapter by making an evaluation for the Jones poly- nomial to feel the need to other types of polynomial link invariants.