An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface

This paper concerns pseudo-classical knots in the non-orientable manifold $\hat{\Sigma} =\Sigma \times [0,1]$, where $\Sigma$ is a non-orientable surface and a knot $K \subset \hat{\Sigma}$ is called pseudo-classical if $K$ is orientation-preserving path in $\hat{\Sigma}$. For this kind of knot we introduce an invariant $\Delta$ that is an analogue of Turaev comultiplication for knots in a thickened orientable surface. As its classical prototype, $\Delta$ takes value in a polynomial algebra generated by homotopy classes of non-contractible loops on $\Sigma$, however, as a ground ring we use some subring of $\mathbb{C}$ instead of $\mathbb{Z}$. Then we define a few homotopy, homology and polynomial invariants, which are consequences of $\Delta$, including an analogue of the affine index polynomial.