Invariants for links in thickened surfaces and its applications

关于三维空间中交错链环的交叉数的Tit猜想是指以下两个猜想：Tc1)如果两个连通、约简、交错的链接图表示等价的(即，环境同位素的)链环，则它们的交叉数相同。TC2)一个连通的、约简的、交错的链接图D的交叉数在表示等同于D的链接图的所有交叉数中将是最小的。这些猜想100多年来一直没有得到解决，直到1987年琼斯多项式出现后，它们才被正解。1996年，N.Kamada做了一项开创性的工作--建立了加厚表面上交替连接的类似结果。在一定的假设下，她向Tc1)展示了一个类似的结果，并猜想在她的论文中可以去掉这个假设。在这个研究项目中，我们肯定地回答了她的猜想。并且，通过应用证明她的猜想的思想，我们建立了一个类似的结果…TC2)，这是寻求与加厚表面的纽结理论中的Tait猜想类似的结果的最终目标，因为TC2)意味着Tc1。我们的这些结果在更一般的表面图和链接图上都比她的论文中的结果更普遍。在这些结果的证明中，我们使用了以下两个基本引理：吞没引理：对于一个曲面上的两个链接图，在一定条件下，它们在曲面上的一个正则邻域可以通过曲面上的同位素变形包含另一个。对偶态引理(加厚曲面版)：对于一个曲面上的链接图D的一对对偶状态，根据每个状态分解所有交叉点得到的两个(不相交)图的连通分量的个数之和不超过一定条件下曲面D的正则邻域的边界分量的个数。这些引理本身，是我们研究中的重要成果。作为与本项目相关的基础性研究，每位研究人员都取得了自己感兴趣的成果。较少

Tait conjectures on crossing numbers of alternating links in the 3-dimensional space means the following two conjectures :TC1) If two connected, reduced and alternating link diagrams represent equivalent (ie, ambient isotopic) links, their crossing numbers will be the same.TC2) The crossing number of a connected, reduced and alternating link diagram D will be minimal among all crossing numbers of link diagrams which represent links equivalent to that D does.These conjectures had been unsolved for more than 100 years untill they were positively solved in 1987 after Jones polynomial appeared. In 1996, N.Kamada made a pioneer-work to establish anologous results for alternating links in thickened surfaces. She showed an anologous result to TC1) under a certain assumption and conjectured that the assumption can be removed in her paper. In this research project, we solved her conjecture positively . And, by applying the idea of the proof of her conjecture, we established an anologous result … More to TC2), which is a final goal to seeking anologous results to Tait conjectures in knot-link theory in thickened surfaces because TC2) implies TC1. Our these results was showed for more general both surfaces and link diagrames than those in her paper. In the proofs of our these results, we use the following two fundamental lemmta :ENGULFING LEMMA : For two link diagrams on a surface, one of their regular neiborhoods in the the surface can contain the other by isotopic deformation in the surface under a certain condition,DUAL STATE LEMMA (thickened surface version) : For a pair of dual states of a link diagram D on a surface, the sum of numbers of connected components of two (no crossing) diagrams obtained from D by resolving all crossings according to each state does not exceed to the number of boundary components of the regular neighhood of D in the surface under a certain condition.These lemmata, themselves, are impotant results in our resaerch. Each invesigator got intereting own results as the fundamental research related to this project. Less