The Unknotting Number: Knots, Surfaces, 3-Manifolds

Abstract

The unknotting number u(K) of a knot K is a natural measure of the knot’s complexity which is easy to define intuitively but has proven difficult to study mathematically. In particular, upper bounds on u(K) are relatively easy to find experimentally, but lower bounds are harder to come by. In this thesis we survey some of the knot invariants and techniques which have been used to give lower bounds on u(K). We focus in particular on the branched double cover of a knot and its use in Lickorish’s obstruction, as well as Scharlemann’s lower bound for the unknotting number of composite knots. This thesis may also serve as a brief, though not entirely self-contained, introduction to knot theory and some of the techniques used therein, for the general mathematical reader at an intermediate-to-advanced undergraduate level.