Jones-Wenzl Idempotents in the Twisted $I$-bundle over the M\"obius band

TL;DR

This paper investigates the properties of Jones-Wenzl idempotents within the Kauffman bracket skein module of the twisted I-bundle over the M"obius band, revealing new algebraic behaviors specific to unorientable surfaces.

Contribution

It introduces the study of Jones-Wenzl idempotents in the skein module of the twisted I-bundle over the M"obius band, highlighting novel algebraic properties distinct from orientable cases.

Findings

Results on Jones-Wenzl idempotents in the skein module of the twisted I-bundle over the M"obius band.

Identification of properties preserved under the I-bundle structure.

Differences from the skein module of the annulus times I.

Abstract

The Jones-Wenzl idempotent plays a vital role in quantum invariants of $3$-manifolds and the colored Jones polynomial; it also serves as a useful tool for simplifying computations and proving theorems in knot theory. The relative Kauffman bracket skein module (RKBSM) for surface $I$-bundles and manifolds with marked boundaries have a well understood algebraic structure due to the work of J. H. Przytycki and T. T. Q. L\^e. It has been well documented that the RKBSM of the $I$-bundle of the annulus and the twisted $I$-bundle over the M\"obius band have distinct algebraic structures coming from the $I$-bundle structures. This paper serves as an introduction to studying the trace of Jones-Wenzl idempotents in the Kauffman bracket skein module (KBSM) of the twisted $I$-bundle of unorientable surfaces. We will give various results on Jones-Wenzl idempotents in the KBSM of the twisted $I$-bundle over the M\"obius band when it is closed through the crosscap of the M\"obius band. We will also uncover analog properties of Jones-Wenzl idempotents in the KBSM of the twisted $I$-bundle over the M\"obius band with the preservation of the $I$-bundle structure that differ from the KBSM of $Ann \times I$.