Comprehensive Research on Topological Graph Theory

拓扑图论综合研究

• 批准号: 05302006
• 负责人: SUZUKI Shinichi
• 金额: $ 3.9万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Co-operative Research (A)
• 财政年份: 1993
• 资助国家: 日本
• 起止时间: 1993 至 1994
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-05302006/
• 关键词: Spatial graphs; Plane graphs; Knots; Links; 3-Manifolds; 結び目理論

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.

1.对于结和链接。引入了各种拓扑等价关系，例如(0)同纯，(1)环境同位素，(2)链路共配，(3)同位素，(5)链路同伦。我们将这些关系推广到图的空间嵌入中。谷山定了各种新的等价关系；即(4)i -等价，(6)弱链同伦，(7)同调，(8)z_2 -同调，并建立如下关系：(0)*(1)*(2)*(4)*(5)*(6)*(7)*(8)；(1) * (3) * (4);(2) <双箭头>(3)。我们研究了空间图在(i)-等价下的各种不变量，特别是K.Taniyam在(5)和(7)下引入了一些新的不变量，Y.Yokota在(1)-不变量下引入了一些新的不变量。T.Kanenobu等研究了Vassiliev型不变量。空间图成为我们小组关注的焦点，许多人尝试将结点和链接的概念和不变量推广到空间图中。例如，Y.Ohyama研究了空间图的一些局部移动，Mohashi-Ohyama-Taniyama根据Yamada多项式的降阶估计了空间图的正则图的最小nm交叉数。S.Kinoshita等研究了若干图上3球分支的分支覆盖空间。森本晃司,T。KOBAYASHI，T等人获得了许多关于隧道结数的结果。小林通过使用书中的演示文稿提出了图形的“标准”空间嵌入。有人研究过这个话题。特别地，T.Otsuki从标准嵌入中选取了“规范”属性，并给出了一些基本属性。

项目成果

Masaharu KOUNO: "On satellite knots" Math.Proc.Camb.Phil.Soc.115. 219-228 (1994)

Masaharu KOUNO：“关于卫星结”Math.Proc.Camb.Phil.Soc.115。

S.Kinoshita :"On the three-fold irregular branched coverings of spatial four-valent graphs and its applications." J.Math.Chem.14. 47-55 (1993)

S.Kinoshita：“关于空间四价图的三重不规则分支覆盖及其应用。”

K.Taniyama :"On embeddings of graphs into R^3." Contemp.Math.164. 239-246 (1994)

K.Taniyama：“关于图到 R^3 的嵌入。”

T.Endo and T.Otsuki :"Notes on spatial representations of graphs." Hokkaido Math.J.23. 383-398 (1994)

T.Endo 和 T.Otsuki：“关于图的空间表示的注释。”

H.Naka :"On the n-fold cyclic branched coverings of a spatial rheta-curves." Kobe J.Math.11. 69-87 (1994)

H.Naka：“关于空间 rheta 曲线的 n 重循环分支覆盖。”