内在纽结图和内在链环图若干问题的研究

The main objects in spatial graph theory are the patterns of graphs embedding in three Euclid space (or three sphere), and there invariants. Knot theory is regarded as a special case of spatial graph theory. Intrinsically knotted graphs and intrinsically linked graphs are hot topics in spatial graph theory during the past few decades. Some remarkable research achievements have been made, but there are still many open problems which are fundamental and important, including the characterizations of the Adams intrinsically knotted graphs. We will study the Adams intrinsically knotted graphs in the following three aspects: Firstly, we will seek for more general Adams intrinsically knotted graphs and uncover the essential features of such intrinsically knotted graphs. Then, the relationship between the general intrinsically knotted graphs and the intrinsically linked graphs will be investigated through graph transformations. Finally, linking number and Conway polynomial will be employed to describe the complexity of intrinsic knotting and intrinsic linking. For an intrinsically complete graph, we will study the change law of the number of vertices, the linking number of a link contained in it, and the second coefficient of the Conway polynomial of a knot contained in it. The project uses a combination of some classical theories, such as knot theory, algebraic topology and graph theory, and therefore it is significant in the study of intrinsically knotted graphs and intrinsically linked graphs. It is beneficial not only to unravel the in-depth underlying properties involved in these two kinds of graphs, but also to make further progress in determining the Adams intrinsically knotted graphs.

空间图理论主要研究图在三维欧式空间（或三维球面）嵌入的方式和不变量，纽结理论可作为其特例。内在纽结图和内在链环图是近年来空间图理论的热点研究课题，已取得了令人瞩目的研究进展，但仍有很多基本且重要的问题尚未解决，其中包括Adams内在纽结图的刻画。本项目主要围绕Adams内在纽结图开展如下研究: （1）寻找更一般类型的Adams内在纽结图，刻画这类内在纽结图的本质特征；（2）通过图变换研究一般的内在纽结图和内在链环图之间的关系；（3）用环绕数、Conway多项式描述内在纽结图和内在链环图的复杂程度，分别探讨内在完全图所含链环的环绕数、内在完全图所含纽结的Conway多项式二次项系数与其顶点数的变化规律。本项目运用纽结理论、代数拓扑、图论等多种理论研究内在纽结图和内在链环图具有重要的意义，有利于深刻揭示这两类图所蕴含的内在性质，同时将在确定Adams内在纽结图问题的研究上取得突破性的进展。

本项目主要运用了纽结理论、代数拓扑、图论等理论，对内在纽结图和内在链环图的若干问题进行了系统地研究。首先，我们得到了较为一般的Adams内在纽结图，进而刻画了Adams内在纽结图的本质；其次，引入图的一些变换，给出了内在纽结图和内在链环图之间某些关系的直观描述；最后，通过计算环绕数和Conway多项式的二次项系数研究内在空间图所含纽结和链环的复杂程度，明确了内在完全图的顶点数对所含链环的环绕数、所含纽结的Conway多项式的二次项系数的影响。本项目的研究结果揭示了内在纽结图和内在链环图蕴含的拓扑性质和几何结构，同时得到了更大的一类Adams内在纽结图，在某种程度上促进了Adams内在纽结图的研究。此外，项目组对本领域的相关热点问题也进行了较为深入的研究，如三维流形的融合，群的亏格谱等，并得到了较好的结果。