扭曲对偶:一个新方向及近期进展

Twuality:A New Direction and Recent Works

胞腔嵌入图(等价地,带子图)的扭曲对偶概念于2012年由Ellis-Monaghan和Moffatt提出,受到了广泛的关注并被应用于拓扑图论、纽结论、拟阵/Delta拟阵和物理学中.本文介绍其起源与发展.首先,我们简单地介绍了纽结论(特别是著名的Jones多项式)、图的Tutte多项式、以及这两个多项式分别在虚链环和带子图上的推广.接着我们回顾了20世纪80年代末建立的这两个多项式之间的关系,然后介绍不同的学者如何在新世纪建立了这两个推广后的多项式之间的新的关系.这些新的关系促使了部分对偶概念在2009年的诞生,统一了这些新的但不同的关系.通过结合Petrie对偶进一步产生了扭曲对偶这一新概念.最后,介绍了我们在带子图扭曲对偶方面的若干研究工作以及它们在Delta拟阵上的推广.

Twisted duality(briefly,twuality)was introduced by Ellis-Monaghan and Moffatt in 2012.It has received ever-increasing attention,and its applications span topological graph theory,knot theory,matroids/delta-matroids,and physics.In this article,we give an account of its origin and its developments.The article is organized as follows.We first give a brief introduction to knot theory,in particular,the celebrated Jones polynomial,then the Tutte polynomial of graphs,and their generalizations to virtual links and ribbon graphs,respectively.We recall the old relation in the late 1980 s between the two polynomials,and explain how different researchers establish the new but different relations between two generalized polynomials in the new century,which,we think,motivate the introduction of partial duality in 2009 to unify the new but different relations.More general new concept twisted dual(briefly,twual)was further introduced by combining the old concept Petrie dual(briefly,Petrial).Finally we report some of our recent works on twuals of ribbon graphs and analogous concepts of delta-matroids.