Study of Knot theory from Geometric Structure

1.内田研究了三桥结和三辫结上三维球面分支的三重不规则分枝覆盖空间，证明了覆盖空间是L(n，1)型和L(n，m)型的透镜空间。(2)如果一个环面纽结有一个三重不规则分支覆盖空间，则它的类型是T(2x，3y)，其中x和y是互素整数。它的覆盖空间是M型(β_1/2x，β_1/x，β_2/Y)的Seifert纤维空间，其中β_1和β_2是2xβ_2+3yβ_1=±1的整数。此外，对于T(2，x)和T(3，x)，他用纽结图证明了这个定理。Ashikaga研究拓扑学和代数几何。(1)描述了关于黎曼曲面族的研究的最新进展。该领域位于拓扑学、代数几何、复分析等相互复杂的边界区域。(2)利用两种互不相同的方法，给出了Dedekind型和的一个公式。一种是初等数论方法，另一种是几何方法。并用该公式重新证明了互易性定律。Torisu给出了(1)具有2-邻接关系的链路的一些例子，他还报道了最近对具有2-邻接关系的2-网桥链路的研究。(2)给出了一个双桥纽结或环S(p，q)2-邻接另一个双桥纽结或环S(r，S)的必要条件。特别地，他证明了：如果平凡纽结或环2-邻接S(p，q)，则S(p，q)是平凡的；如果S(p，q)2-邻接于其镜像，则S(p，q)是双面的；对于素数p，如果S(p，q)2-邻接S(r，S)，则S(p，q)=S(r，S)或S(r，S)=S(1，0)。(3)给出了一个强1-平凡Montesinos纽结的Off-Lain族，并证明了如果关于Seifert手术的一个著名猜想成立，则该族包含所有强1-平凡的Montesinos纽结。

1. Uchida studies (1) the three-fold irregular branched covering space of three-dimensional sphere branched over three-bridge knot and three-braid knot And he showed that the covering space is a lens space of type L (n, 1) and L (n, m), respectively. (2) lf a torus knot has a three-fold irregular branched covering space, then its type is T (2x, 3y), where x and y are co-prime integers. Its covering space is a Seifert fibered space of type M (β_1/2x, β_1/x, β_2/Y), where β_1 and β_2 are integers with 2xβ_2+3yβ_1=±1. Moreover for T (2, x) and T (3, x), he proved this theorem by using a knot diagram.2. Ashikaga studies topology and algebraic geometry. (1) He describes the recent development of study of degenerate families of Riemann surfaces. This field is located at the boundary area where topology, algebraic geometry complex analysis and others get complicated each other. (2) He propose a certain formula of Dedekind sum by using two mutually distinct methods. The one is an elementary number theoretic method and the other is a geometric method. Moreover he re-prove the reciprocity law by the formula.3. Torisu gives (1) some examples of links having 2-adjacency relation and he also reports a recent study of 2-bridge links with 2-adjacency relation. (2) He gives a necessary condition for a two-bridge knot or link S (p, q) to be 2-adjacent to another two-bridge knot or link S (r, s). In particular he shows that if the trivial knot or link is 2-adjacent to S (p, q), then S (p, q) is trivial, that if S (p, q) is 2-adjacent to its mirror image, then S (p, q) is amphicheiral, and that for a prime integer p, if S (p, q) is 2-adjacent to S (r, s), then S (p, q)=S (r, s) or S (r, s)=S (1, 0). (3) He present a off Lain family of strongly 1-trivial Montesinos knots, and show that if a well known conjecture on Seifert surgery is valid, then the family contains all strongly 1-trivial Montesinos knots.