Geometric Knot Theory

Abstract

This thesis provides an exposition to some geometric aspects of knot theory. We discuss certain numerical invariants of knots which are geometric in nature. Many of them are defined as the minimum value of some quantity over all the directions taken over one particular diagram or configuration and then minimizing this over all possible configurations. Crossing index, unknotting index and bridge index are some of the examples of such invariants. Later some invariants were defined by first taking the maximum value of these quantities over all the directions taken over one particular diagram or configuration and then minimizing this over all possible configurations. They were termed as superinvariants. Superbridge index, supercrossing index and superunknotting index are studied lately. All these invariants are very difficult to compute. Certain parametrizations are used to obtain some bounds for these invariants. Polygonal representation for knots has been instrumental in PL category. Similarly polynomial representation plays an important role in smooth category. In this thesis we also study the topology of the spaces of polygonal knots as well as polynomial knots. We discuss some applications of these spaces at the end.