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