Comprehensive Research on Topological Graph Theory

1.用于结和链接。引入了各种拓扑等价关系，例如，（0）同胚，（1）环境同位素，（2）链接共配，（3）同位素，（5）链接同伦。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；即(4)I-等价性、(6)弱连接同伦性、(7)同源性和(8)Z_2-同伦性，并建立以下关系：(0)*(1)*(2)*(4)*(5)*(6)*(7)*(8)； (1)*(3)*(4) ; (2) <双箭头> (3)。我们研究了每个 i 的 (i) 等价性下的空间图的各种不变量。特别是，K.Taniyam 在 (5) 和 (7) 下引入了一些新的不变量，而 Y.Yokota 则引入了一些 (1)-不变量。 T.Kanenobu 等人研究了 Vassiliev 型不变量2。空间图成为我们小组关注的中心，许多人试图将结和链接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多项式的约化次数估计了空间图正则图的最小交叉数。 S.Kinoshita 等人研究了在一些图上分支的 3 球体的分支覆盖空间。 MORIMOTO,T.KOBAYASHI,T等人在隧道结数方面取得了许多成果。3.K.Kobayashi利用书中的介绍提出了图的“标准”空间嵌入。有人研究过这个话题。特别是，T.Otsuki 从标准嵌入中提取了“规范”嵌入，并给出了一些基本属性。

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.