| Research Abstract |
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.
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