An Analog of the Kauffman Bracket Polynomial for Knots in the Non-orientable Thickening of a Non-orientable Surface

We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable 3-manifold of the form (non-orientable surface) \(\times \) $\times$ [0, 1]. For these knots, we propose an analog of the Kauffman bracket polynomial. The construction of this polynomial closely mirrors the classical version, with key differences in the definitions of the sign of a crossing and the positive/negative smoothing of a crossing. We prove that this polynomial is an isotopy invariant of pseudo-classical knots and demonstrate that it is independent of the classical Kauffman bracket polynomial for knots in the thickened orientable surface, which is the orientable double cover of the non-orientable surface under consideration.