Comprehensive Research on Topological Graph Theory

点击翻译按钮获取中文摘要

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.