A proof of the shadow formula for the SU(2)-Reshetikhin-Turaev-Witten invariant

TL;DR

This paper provides a concise proof of Turaev's shadow formula for the SU(2)-Reshetikhin-Turaev-Witten invariants, connecting topological invariants with skein theory techniques.

Contribution

It offers the first short proof of the shadow formula using skein theory, extending its applicability to colored framed graphs in 3-manifolds.

Findings

Validates the shadow formula for a broad class of 3-manifolds and graphs

Establishes a skein-theoretic approach to topological invariants

Simplifies the computation of SU(2)-Reshetikhin-Turaev-Witten invariants

Abstract

Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly empty) colored framed trivalent graph (for instance, a framed knot or link).