MATHEMATICS
Papers & Preprints[0] Diagrammatic understandings of the relationship between the colored Kauffman bracket of a knot
and geometric properties of the knot complement,
Ph.D thesis, Kyushu University, Japan (2004).
[1] A diagrammatic construction of the (sl(N,C),\rho)-weight system,
Interdisciplinary Information Sciences, Vol.9, No.1, pp.43-51 (2003).
[2] Efficient formula of the colored Kauffman brackets,
Lobachevskii Journal of Mathematics, Vol.16, pp.71-78 (2004).
[3] Computing the A-polynomial using noncommutative methods,
Journal of Knot Theory and Its Ramifications, Vol.14, No.6, pp.735-749 (2005).
[4] Some results about the Kauffman bracket skein module of the twist knot exterior,
(joint work with R. Gelca),
Journal of Knot Theory and Its Ramifications, Vol. 15, No. 8, pp.1095-1106 (2006).
[5] An exposition of the sl(N,C)-weight system,
(joint work with T. Takamuki),
Journal of Lie theory, Vol.17 No.2, pp.263-281 (2007).
[6] Finiteness of a section of the SL(2,C)-character variety of knot groups,
Kobe Journal of Mathematics, Vol. 24, no. 2 (2007).
The methods used in this paper can be applied to small knots.
Paper [6] is dedicated to the memory of one of my supervisors, Professor Xiao-Song Lin, who passed away in 2007.
A main result in this paper was generalized to SL(n,C) by Hans U. Boden and Stefan Friedl
(refer to their preprints 1 and 2).
[7] Background of the existence of multi-variable link invariants,
(joint work with K. Hamai),
Kyungpook Mathematical Journal, Vol.48, pp.233-240 (2008).
[8] On a behavior of a slice of the SL_2(C)-character variety of a knot group under the connected sum,
Topology and its Applications, Vol. 157, pp.182-187 (2010).
[9] On the character rings of twist knots,
Bulletin of the Korean Mathematical Society, Vol. 48, No. 3, pp.469-474 (2011).
[10] On the geometry of the slice of trace-free SL(2,C)-characters of a knot group,
(joint work with Yoshikazu Yamaguchi),
Mathematische Annalen, Vol. 354, pp.967-1002 (2012) (DOI: 10.1007/s00208-011-0754-0).
An old version is available at http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.0714v3.pdf.
Note that this paper is a revised version of the paper entitled "On the geometry of a certain slice of the character variety of a knot group" (the title has been changed!).
[11] On minimal elements for a partial order of prime knots,
Topology and its Applications, Vol. 159, pp.1059-1063 (2012).
This explains an application of Paper [9]. More precisely, applying Chebyshev polynomials, we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL(2,C)-character variety of twist knots in infinitely many cases. The PSL(2,C) version of this fact is mentioned by Michel Boileau and Steve Boyer in their preprint On character varieties, sets of discrete characters, and non-zero degree maps (see Example 3.3 on Page 21). The irreducibility, combined with a result in the paper of Boileau et al Simon's conjecture for 2-bridge knots, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups.
We also give a straightforward proof of the result of Boileau et al.
[12] On the trace-free characters,
RIMS Kokyuroku, Vol. 1836, "Representation spaces, twisted topological invariants and geometric structures of 3-manifolds", pp.110-123 (2013)
[13] Some families of minimal elements for a partial ordering on prime knots,
(joint work with Anh T. Tran),
Osaka Journal of Mathematics, Vol. 53, Ｎo.4. (2016)
Available at http://arxiv.org/abs/1301.0138.
We show that certain types of 2-bridge knots including twist knots are minimal elements for a partial ordering on the set of prime knots. To prove the result, we apply presentations of the character varieties using Chebyshev polynomials and give a criterion for irreducibility of a polynomial. This gives us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.
(There are several things to be corrected in the manuscript on ArXiv.)
[14] On minimality of 2-bridge knots,
(joint work with Masaaki Suzuki and Anh T. Tran),
International Journal of Mathematics, Vol. 28 (2017)
[15] Algebraic varieties via a filtration of the Kauffman bracket skein module and knot contact homology ,
Topology and its applications (2019), https://doi.org/10.1016/j.topol.2019.06.024.
This paper, based on the preprint written in 2007, explains a research on the character varieties and the Kauffman bracket skein module (KBSM) of knot exteriors, which has been done by the author in his stay at the University of California, Riverside, 2004-2006.
A survey of this paper from a representation theoretical viewpoint concerned with the Casson and the Casson-Lin
invariants is in Notes & Reports [6] (Japanese version only).
This paper is dedicated to Professor Xiao-Song Lin.
The results in this paper have been generalized to Papers [10,16,17,18].
[16] Trace-free characters and abelian knot contact homology I,
Available at https://arxiv.org/pdf/1708.00851.pdf.
[17] Trace-free characters and abelian knot contact homology II,
(joint work with Shinnosuke Suzuki),
Available at https://arxiv.org/pdf/1708.00874.pdf.
[18] Ghost characters and character varieties of 2-fold branched covers,
(joint work with Shinnosuke Suzuki),
Available at https://arxiv.org/pdf/1708.01511.pdf.
In Preparation[1]
Ghost characters of knots.[2]
Lecture Notes: Knots, Operations on them and Their Invariants
(with T. Takamuki and H. Nishihara),now in preparation for publication.
This is a lecturenotes of the series of Dylan Thurston's talks at Research Institute for Mathematical Sciences (RIMS), Kyoto, Japan (fall, 2001).
Recent Talks
[1] [scheduled] Sept. 2019: TBA,
International workshop ``Knots in Tsushima 2019'', Tsushima, Japan.
[2] Dec. 8, 2018: On some geometric properties of the SL(2,C)-ghost characters of a knot,
International workshop ``2018 Ryukyu Knot Seminar'', Okinawa, Japan.
[3] Nov. 14, 2017: Trace-free characters and abelian knot contact homology,
The 2nd Pan Pacific International Conference on Topology and Applications, Busan, Korea.
[4] Feb. 13, 2017: Ghost characters, character varieties and abelian knot contact homology,
International conference ``the 12th East Asian School of Knots and Related Topics'', Tokyo, Japan.
Notes & Reports[1] Recursive formula of the colored Jones polynomial for the figure-eight knot with 0-framing,
unpublished (January, 2003).
This note pointed out some errors in the earlier version of the paper of Gelca & Sain
``The computation of the non-commutative generalization of the A-polynomial of the figure-eight knot''
and corrected them.
[2] The minimal relation in the Kauffman bracket skein module of the m-twist knot,
Proceedings of the East Asian School of Knots, Links, and Related Topics, Seoul, Korea (2004).
Another version of this report was published in Proceedings of Intelligence in Low-Dimentional Topology, Syodo-shima (2003).
[3] Some results about the Kauffman bracket skein module of the twist knot exterior (Joint work with R. Gelca),
This report is a part of Paper [4] including the contents of the talk which I gave at First KOOK Seminar International
for Knot Theory and Related Topics, Awaji-shima (2004).
[4] Notes on the Kauffman bracket skein module of a handlebody,
Proceedings of the second East Asian School of Knots and Related Topics in Geometric Topology, Dalian, China (2005).
Notice: I made some mistakes in this report.
[5] Algebraic equations and knot invariants,
Proceedings of Topology of knots IX (2006), Japan. This is a report version of Paper [11].
Notice: There were some mistakes in this report.
[6] Character varietyの断面から誘導される代数多様体族とknot contact homology (in Japanese),
This is a revision of the manuscript in Proceedings of the 54th Topology Symposium, Aizu, Japan (2007).
[7] 指標代数多様体の断面の既約成分数について (in Japanese),
Proceedings of "結び目の数学" (2009), Japan.
This has been completely solved by Corollary 7.17 on page 64 in Kronheimer and Mrowka: Knots, Sutures and excision.
[8] On low-dimensional topology and knot theory (in Japanese),
Res. Rep. Fac. Sci. Technol. Meijo Univ. No. 49 (2009), 13-19.
Lecturenotes[1] The Frohman-Gelca-Lofaro theory (by K. Ichihara, in Japanese),
notes of my talk in "Topology Young Seminar in Kansai" (April, 2003).
[2] Winter Workshop 2008 報告集（レジュメ集） (in Japanese),
This is a report of the conference "Winter Workshop 2008" (Feb., 2008).
Data (files used in my talks)[1] An approach to the A-polynomial of (2,2p+1)-torus knots from the Frohman-Gelca-Lofaro theory,
slides of a talk given in the conference "The 10th Japan-Korea School of Knots and Links", Tokyo (2003).
[2] SL(2,C)-metabelian representations and algebraic varieties: from SL(2,C)-representations to knot contact homology,
slide file of my talk in Topology seminor at Tokyo Institute of Technology (April 25, 2007) and Waseda University
(April 28, 2007).