带子图、纽结和不变量

Knots was originally an object of topology, but appeared gradually in the textbooks of graph theory and combinatorics since 1990s. A ribbon graph is a surface with boundary and graphic structure on it, which is a new topic in graph theory. Its boundary consists of knots, its edges correspond to ribbons. We shall study:.1.Ribbon graphs and twisted duality. Twisted duality is similar to graph transformation. Classical checkerboard graph and Seifert graph in knot theory are twisted dual to each other. We shall consider: (1) given a property, under which conditions, its twisted duals will possess such a property? (2) Given a ribbon graph, how its invariants will change under twisted duality? .2. knot invariants using planar graphs or ribbon grpahs as tools, hoping to settle or make progress in several open problems. For examle: (1) since the introduction of the Jones polynomial, there is a strong desire to have a geometric or topological interpretation for it and it is still open by now. (2) the proof of independence of edge ordering of the combinatorial model of the HOMFLY polynomial introduced by famous combinatorist F. Jaeger. They both have close relations with the Tutte polynomial..We hope through this project we could train some students studying knots and develop some new directions in the field of graph theory of our country.

纽结原属拓扑学，自上世纪90年代开始出现在图论和组合论的部分教科书中。带子图是带有图结构的有边界曲面，其边界是纽结，边对应带子。以纽结论为背景，带子图成为最近几年来的一个研究热点。本项目拟研究：.1. 带子图与扭曲对偶变换。纽结论中经典的棋盘图和Seifert图可以通过扭曲变换相互转化。将考虑：（1）给定某个抽象图的或拓扑的性质，在什么条件下，一个带子图的扭曲对偶具备上述性质？（2）给定一个带子图，研究带子图不变量在扭曲对偶下的变化情况。.2. 以平面图或高亏格带子图为工具研究纽结不变量，希望能推进解决一些公开的问题。例如：（1）Jones多项式系数的几何意义是受拓扑学家关注至今仍不清楚的问题。（2）组合学家Jaeger引入的HOMFLY多项式的组合模型的不依赖于边序的证明问题。这两个问题都与Tutte多项式有关。.我们同时希望在国内图论领域培养纽结研究队伍，发展图论研究的新方向。

纽结原属拓扑学，自上世纪90年代开始出现在图论和组合论的部分教科书中。带子图是带有图结构的有边界曲面，其边界是纽结，边对应带子。以纽结论为背景，带子图成为最近几年来的一个研究热点。本项目研究了：.1..带子图与扭曲对偶变换。.主要解决了任意一个带子图的部分对偶中欧拉图和二部图的刻画问题和任意一个带子图的扭曲对偶中正则的可棋盘染色的带子图的刻画问题。.2..图和纽结不变量。.主要研究了Jones多项式和Tutte多项式的极端项系数的组合意义；将纽结论中的skein关系与拓扑Tutte多项式结合，将空间图的Yamada多项式推广到虚空间图上，它可以成功捕捉到虚纽结的“虚”性；得到了一个简单的一般图的线图的生成树计数公式。..以上研究成果得到了国内外同行的认可，在国际著名期刊发表或接受发表，有些研究方法（如生成树计数的电网络等价方法）得到国内外同行的应用和进一步发展，受邀参加国内国际学术会议并做报告10余次。四年来，共培养博士生3名，硕士生2名，指导访问学者1名；目前正在培养博士生6名，硕士生10名，指导访问学者1名。