Knot Invariants and Topological Quantum Field Theory

TL;DR

This paper introduces knot theory and its connection to topological quantum field theory, detailing calculations of knot invariants via Chern-Simons theory, and discusses recent advances including categorification and computational tools.

Contribution

It provides detailed calculations of knot invariants from Chern-Simons theory, introduces new formulas for superpolynomials, and develops a MATHEMATICA program for computing $6j$-symbols.

Findings

Explicit calculation of Chern-Simons propagator

Derivation of topological and knot invariants from $U(1)$ theory

Development of a computational tool for $6j$-symbols

Abstract

An elementary introduction to knot theory and its link to quantum field theory is presented with an intention to provide details of some basic calculations in the subject, which are not easily found in texts. Study of Chern-Simons theory with gauge group $\mathcal{G}$, along with the Wilson lines carrying some representation is explained in generality, and a vital calculation of the Chern-Simons propagator is done. Explicit calculation for $U(1)$ Chern-Simons theory is presented, which leads to the topological invariants, and finally to knot invariants. Further, using this result along with the Gauss linking number formula, the expectation value of Wilson loops are calculated. Colored knot invariants are also discussed along with more advanced knot invariants which are obtained using Homology theory, i.e., categorification of Jones and HOMFLY polynomials. Various knot invariants for $SU(N)$ gauge group are also introduced, along with a brief introduction to A-polynomials and super A-polynomials. Recent developments in the field are explored, and we discuss a conjectured formula for colored superpolynomials, closed-form expression for HOMFLY polynomials, and conjectured expression for $6j$ symbol for $U_q(\mathfrak{sl}_N)$ for multiplicity free case. Also, a MATHEMATICA program based on the conjectured formula had been developed, which can compute the $6j$-symbols and the desired duality matrices which are needed to use the closed-form expression for HOMFLY polynomials.