Comprehensive Research on Topological Graph Theory

1.对于结和链接。引入了各种拓扑等价关系，例如（0）同伦，（1）环合痕，（2）链配边，（3）合痕，（5）链同伦。我们将这些关系推广到图的空间嵌入。K.Taniyama定义了各种新的等价关系：（4）I-等价，（6）弱连接同伦，（7）同调，（8）Z_2-同调，并建立了如下关系：（0）*（1）*（2）*（4）*（5）*（6）*（7）*（8）;（1）*（3）*（4）;（2）<double arrow>（3）。研究了空间图在对任意i（i）-等价下的各种不变量.特别地，K.Taniyam在（5）和（7）下引入了一些新的不变量，Y.Yokota引入了一些（1）-不变量。Vassiliev型不变量是由T.Kanenobu等人研究的。2.空间图成为我们小组关注的焦点，许多人试图将纽结和链环的概念和不变量推广和/或扩展到空间图。例如，Y.Ohyama研究了空间图的一些局部移动，Mohashi-Ohyama-Taniyama根据Yamada多项式的约化度估计了空间图的正则图的最小交叉数。S.Kinoshita研究了某些图上3-球面分支的分支覆盖空间。et.al 3.K.小林利用书中的介绍提出了图的“标准”空间嵌入。有人研究过这个课题。特别地，T.大月从标准嵌入中挑选了“规范”嵌入，并给出了一些基本性质。

1.For Knots and links. various topological equivalence relations are introduced, for example, (0)homeomorhic, (1)ambient isotopy, (2)link cobordism, (3)isotopy, (5)link homotopy. We generalized these relations for spatial embeddings of a graph. K.Taniyama defined various new equivalence relations ; that is, (4)I-equivalence, (6)weak link homotopy, (7)homology and (8)Z_2-homology, and establishes the following relation : (0)*(1)*(2)*(4)*(5)*(6)*(7)*(8) ; (1)*(3)*(4) ; (2) <double arrow> (3). We studied various invariants of spatial graphs under (i)-equivalence for every i. In particular, K.Taniyam introduced some new invariants under (5)and(7), and Y.Yokota some(1)-invariants. The Vassiliev type invariants were studied by T.Kanenobu et.al.2.Spatial graphs became the center of attention in our group, and many people made attempt to generalize and/or extend the concepts and invariants of knots and links to spatial graphs. For example, Y.Ohyama studied some local moves for spatial graphs, and Mohashi-Ohyama-Taniyama estimated the minimnm crossing number of regular diagrams of spatial graphs in terms of the reduced degree of the Yamada polynomial. S.Kinoshita et.al studied the branched covering spaces of the 3-sphere branched over some graphs. MORIMOTO,T.KOBAYASHI,T et al obtained many results on the tunnel number of knots.3.K.Kobayashi proposed "standard" spatial embeddings of graphs by using the book presentations. and some one studied on this topics. In particular, T.Otsuki picked up "canonical" ones from the standard embeddings, and gave some fundamental properties.