Kauffman bracket skein module of the connected sum of two projective spaces

TL;DR

This paper introduces diagrams and moves for links in a twisted bundle over unorientable surfaces, computing the Kauffman Bracket Skein Module of the connected sum of two projective spaces, revealing torsion, and re-computing known skein modules.

Contribution

It provides a new diagrammatic approach for links in twisted bundles and computes the KBSM for complex 3-manifolds, including the connected sum of projective spaces.

Findings

The KBSM of the connected sum of two projective spaces has torsion.

New diagrams and Reidemeister moves for links in twisted S^1-bundles.

Re-computation of KBSM for S^1 x S^2 and lens spaces L(p,1).

Abstract

Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBSM of S^1 x S^2 and the lens spaces L(p,1).