Torus Knots and the Chern-Simons path integral: a rigorous treatment

TL;DR

This paper rigorously derives a formula for quantum invariants of torus knots in S^2 x S^1 from the Chern-Simons path integral, confirming the classical Rosso-Jones formula through surgery techniques.

Contribution

It provides a rigorous derivation of the Rosso-Jones formula for torus knots in a new manifold using Chern-Simons theory and surgery methods.

Findings

Derived a S^2 x S^1-analogue of the Rosso-Jones formula.

Verified the classical Rosso-Jones formula via surgery operations.

Connected quantum invariants in different 3-manifolds through rigorous path integral methods.

Abstract

In 1993 Rosso and Jones computed for every simple, complex Lie algebra g_C and every colored torus knot in S^3 the value of the corresponding U_q(g_C)-quantum invariant by using the machinery of quantum groups. In the present paper we derive a S^2 x S^1-analogue of the Rosso-Jones formula (for colored torus ribbon knots) directly from a rigorous realization of the corresponding (gauge fixed) Chern-Simons path integral. In order to compare the explicit expressions obtained for torus knots in S^2 x S^1 with those for torus knots in S^3 one can perform a suitable surgery operation. By doing so we verify that the original Rosso-Jones formula is indeed recovered for every g_C.