A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications

TL;DR

This paper introduces a geometric formula for Witten-Reshetikhin-Turaev quantum invariants, analyzes their asymptotics for certain 3-manifolds, and derives knot characterization results based on quantum invariants.

Contribution

It provides a new geometric construction of boundary states and a formula for quantum invariants, linking topology, quantum invariants, and knot theory.

Findings

Quantum invariants differ for knots with irreducible SU(2) representations.

Knots with the same colored Jones polynomials as the unknot are necessarily the unknot.

Asymptotic analysis reveals distinctions in quantum invariants for specific 3-manifolds.

Abstract

We provide a geometric construction of the boundary states for handlebodies which we in turn use to give a geometric formula for the Witten-Reshetikhin-Turaev quantum invariants. We then analyze the asymptotics of this invariant in the special case of a three manifold given by 1-surgery on a knot and we show that if the knot has an irreducible representation of its fundamental group into SU(2), then its quantum invariant cannot equal those of the three sphere. From this we conclude that if a knot has the same colored Jones polynomials as the unknot, it must be the unknot.