Winding parity projection and embedding of virtual knots

It is known that knots in the product space of an oriented surface $S_{g}$ and the circle $S^{1}$ can be presented by virtual diagrams with decorations up to local moves. By using the first homology of $S^{1}$ one can define a parity-like invariant for knots in $S_{g} \times S^{1}$, which is called a winding parity. In this talk, we define a projection of knots in $S_{g}\times S^{1}$ with degree $0$ onto knots with zero winding parity for all crossings. By using the projection, we prove that virtual knots are “almost” embedded into knots in $S_{g} \times S^{1}$.