Links, Quantum Groups, and TQFT's

TL;DR

This paper explores the construction of knot invariants like the Jones polynomial via quantum groups and tangle functors, and develops related 3-manifold invariants and topological quantum field theories.

Contribution

It explicitly constructs the quantum group U_q(sl_2) and links it to knot invariants and TQFTs, providing a unified formalism for these concepts.

Findings

Constructed the Jones polynomial and Kauffman bracket

Developed quantum group and tangle functor formalisms

Built 3-manifold invariants and TQFTs from link invariants

Abstract

The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial, is constructed explicitly. The $3$-manifold invariants and the axiomatic topological quantum field theories which arise from these link invariants at certain values of the parameter are constructed.