Journal of Knot Theory and Its Ramifications 26 1750018-1-1750018-23 Feb 2017 [Refereed]

For a link with zero determinants, a ℤ-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial ℤ-coloring a ℤ-colorable link. The minimal coloring number of a ℤ-colorable link is the mini...

A lower bound on minimal number of colors for links

Kazuhiro Ichihara, Eri Matsudo

Kobe Journal of Mathematics 33(1-2) 53-60 Dec 2016 [Refereed]

We show that the minimal number of colors for all effective n-colorings of a link with non-zero determinant is at least 1+log_2 n.

Communications in Analysis and Geometry 24(2) 337-377 Jun 2016 [Refereed]

We give a complete classification of exceptional surgeries on hyperbolic alternating knots in the 3-sphere. As an appendix, we also show that the Montesinos knots M(−1/2,2/5,1/(2q+1)) with q > 4 have no non-trivial exceptional surgeries. This give...

Experimental Mathematics 25(1) 66-78 Jan 2016 [Refereed]

For a given cusped 3-manifold admitting an ideal triangulation, we describe a method to rigorously prove that either the manifold or a filling of the manifold admits a complete hyperbolic structure via verified computer calculations. Central to o...

Kazuhiro Ichihara, Makoto Mori, and Ken-ichi Yoshida

RIMS Kôkyûroku 1960 1-17 Aug 2015

We consider the random link, which is defined as the closures of braids obtained from random walks on the braid groups. For random links, the expected value for the number of components were calculated by Jiming Ma. In this article, we report on t...

Journal of Osaka Sangyo University. Natural sciences 117 33-43 Feb 2006

In this research note, a number of bounds on boundary slopes of essential surfaces embedded or immersed in 3-manifolds are presented. Also reports on computer- aided experiments, concerning embedded boundary slopes for Montesinos knots, are included.

The well-known L-space conjecture saids that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. One of the known approaches to the conjecture is by using Dehn surgery. In this talk, ...

Invariants of 3-manifolds related to the Casson invariant 26 Jan 2017 Research Institute for Mathematical Sciences, Kyoto University

I will talk about two generalizations of the Casson invariant, and their applications to the cosmetic surgery conjecture on knots. One is the SL(2,C) version of the Casson invariant originally introduced by Curtis, and the other is the degree 2 pa...

Fundamental Groups, Representations and Geometric Structures in 3-Manifold Topology 22 Nov 2016

I will talk about the SL(2, C) Casson invariant for 3-manifolds, and its applications to the cosmetic surgery problem for knots in the 3- sphere. In particular, in terms of boundary slopes, a condition for knots to admit no cosmetic surgeries will...

It was conjectured by L.M. Lopez that small knots always exist in closed small 3-manifolds. In this talk, I will talk about the current status of this conjecture, and show that every lens space contains a hyperbolic small knot.