On winding parities of crossings of knots in $S_{g} \times S^{1}$

For knots in $S_{g} \times S^{1}$, where $S_{g}$ is an oriented surface of genus $g$, one of important information is “how many times a half of a crossing turns around $S^{1}$, and we call it winding parity of a crossing. In this talk we extend this notion more generally and discuss its meaning. This is an ongoing work.