Links, Quantum Groups and TQFTs

Abstract

This thesis aims to provide a detailed understanding of quantum invariants of knots and links. We present them as the link invariants derived from a representation of some quantum group. The most celebrated invariant, the Jones polynomial, is shown to be obtained from the fundamental representation of the quantum group Uh(sl2(C)), making it an example of a quantum invariant. In this thesis, we focus on the computation of the Jones polynomial. We provide a closed-form expression for the Jones polynomial of the weaving links W(3, m) and observe some patterns in the coefficients of the Jones polynomial of weaving links W(4, m). We emphasise the role of quantum groups as a general machinery for generating link invariants. It is known that for every semi-simple lie algebra g, one can obtain a quantum group Uh(g) and define a new link invariant for each of its representations. We also discuss a general approach for constructing link invariants called ‘Topological Quantum Field Theories (TQFTs)’. Quantum invariants are examples of 1-dimensional TQFTs. This thesis discusses the converse: every 1-dimensional TQFT arises from a representation of some quantum group. Looking at quantum invariants as 1-dimensional TQFTs allows us to generalise further and study invariants arising from n-dimensional TQFTs. Thus, one has a much broader definition of a quantum invariant, namely those arising from an n-dimensional TQFT.