In knot theory not only classical knots, which are embedded circles in S^{3} up to isotopy, but also knots in other 3-manifolds are interesting for mathematicians. In particular, virtual knots, which are knots in thickened surface $S_{g} \times [0,1]$ with an orientable surface $S_{g}$ of genus $g$, are studied and they provide interesting properties.
In this talk, we will talk about knots in $S_{g} \times S^{1}$ where $S_{g}$ is an oriented surface of genus $g$. We introduce basic notions and properties for them. In particular, for knots in $S_{g} \times S^{1}$ 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. We extend this notion more generally and introduce a topological model, which provide a universal winding parity.
KIM SEONGJEONG博士毕业于莫斯科国立大学,现就职于吉林大学数学学院。研究方向为低维拓扑,纽结理论。已发表SCI论文12篇,纽结理论书籍1部,获得青年基金项目1项。