Jones-Wenzl Idempotents in the Twisted $I$-bundle over the Möbius band

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. Lê. It has been well documented that the RKBSM of the $I$-bundle of the annulus and the twisted $I$-bundle over the Möbius 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 Möbius band when it is closed through the crosscap of the Möbius band. We will also uncover analog properties of Jones-Wenzl idempotents in the KBSM of the twisted $I$-bundle over the Möbius band with the preservation of the $I$-bundle structure that differ from the KBSM of $Ann\times I$.