論文 / 著書情報
Article / Book Information
題目(和文)

双曲体積，ファイバーの通約性，そして例外的手術．理論と計算，双方向からの研究

Title(English)

Hyperbolic Volume, Fibered Commensurability, and Exceptional
Surgeries, Theory v.s. Computation

著者(和文)

正井秀俊

Author(English)

Hidetoshi Masai

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第9421号,授与年月日:2014年3月26日,
学位の種別:課程博士,
審査員:小島 定吉,渡辺 治,梅原雅顕,西畑伸也,寺嶋Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第9421号,
Conferred date:2014/3/26,
Degree Type:Course doctor,Examiner:,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

論文要旨

Type(English)

Summary

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郁二

（博士課程）Doctoral Program

論	 文	 要	 旨	 ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ	 
専攻：	 
Department of

数理・計算科学	 

学生氏名：	 Studentʼs Name

正井 秀俊	 

専攻	 	 

申請学位（専攻分野）：	  博士	 
Academic Degree Requested	 	 

指導教員（主）：	 
Academic Advisor(main)	 

指導教員（副）：	 
Academic Advisor(sub)Doctor of（	  理学	  ）	 小島	  定吉	 	 	 要旨（英文 800 語程度）	 
Thesis Summary （approx.800 English Words ）	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 A hyperbolic 3-manifold is a Riemannian manifold of constant sectional curvature −1. We mainly consider
hyperbolic 3-manifolds of finite volume. By the Mostow-Prasad rigidity theorem, two such manifolds areisometric if and only if they are homeomorphic. Hence we may study hyperbolic geometry via topology
and vice versa. Further, thanks to the Mostow-Prasad rigidity, computers can be very useful tools for thestudy of hyperbolic 3-manifolds. In this thesis, we study hyperbolic 3-manifolds both theoretically andcomputationally.
In the theoretical part, we study hyperbolic volumes, and fibered commensurability. We first discuss avolume preserving move which relates hyperbolic graphs and links. By using this move, we give a method
to compute the best possible upper bounds for volumes of hyperbolic link complements in terms of thetwist numbers. We compute approximate values of the best possible upper bounds for the case where twist
numbers are less than 10 by using the computer programs called Orb and plantri. Next, we discuss thenumber N (v) of hyperbolic 3-manifolds of volume v. By a work of Jørgensen and Thurston, the number N (v)is known to be finite for all v ∈ R. We first investigated censuses on SnapPea and computed (approximate)volumes. This experiments suggest that i) supw≤v N (w) grows exponentially as v goes to infinity, and ii)
there are infinitely many v with N (v) = 1. We show that these are actually the case. As the final work about
hyperbolic volumes, we compare two formulae by Bridgeman-Kahn and Calegari that compute the volumeof a given compact hyperbolic n-manifold M with totally geodesic boundary in terms of orthospectrum. The
orthospectrum of M is the set of length of orthogeodesics, geodesics perpendicular to the boundary ∂M atboth ends. The two formulae are derived by means of different decompositions of the unit tangent bundle.
Since both formulae hold for every such hyperbolic manifold, it is natural to ask if their formulae coincide.
For n = 2, Calegari showed that they are identical. We verify that their formulae coincide for any n ≥ 2.
Both Bridgeman-Kahn and Calegari showed that their formulae can be expressed by certain integrals which
are complicated. We derived a simple expression of their formulae for the case where n = 3; for a pieceassociated to an orthospectrum of length l, it will be 2π(l+1). We then consider fibered commensurability ofe2l −1fibrations on a hyperbolic 3-manifold. The notion of fibered commensurability is introduced by CalegariSun-Wang. We first give a necessary condition for manifolds to have non-symmetric but commensurable
fibrations. Then, we also show that for any n ∈ N, there exists a hyperbolic 3-manifolds with at least nmutually non-symmetric but commensurable fibrations.
We now turn to a computational part of this thesis. We first developed the package hikmot that enables
us to prove the hyperbolicity of a given triangulated 3-manifold. Since, to prove the hyperbolicity, wedeal with equations with complex variables, so-called gluing equations, the notion of interval arithmetic
plays a key role. Note that computers can only deal with approximate values. Instead of dealing withapproximate values, in interval arithmetic, we compute intervals that contain rigorous values. Thus wecan prove inequalities. To solve a gluing equation, we use Newton’s method. There are several theorems
that give sufficient conditions, which are inequalities, for the convergence of Newton’s method. Putting
those theorems and interval arithmetic together, we can prove that there is a solution for a gluing equationand hence the hyperbolicity of a manifold. By using hikmot we prove hyperbolicity of all manifolds inseveral censuses on SnapPy. Further, we use hikmot together with a python code based on the codes byMartelli-Petronio-Roukema to classify all the exceptional surgeries along alternating knots. Thurston has
proved that for a given hyperbolic knot, all but finitely many surgeries give hyperbolic manifolds. We callthose finitely many surgeries exceptional. The codes by Martelli-Petronio-Roukema enumerate exceptional
surgeries by using floating point arithmetic. We modified their codes so that each computation is done byinterval arithmetic. Thanks to several known works, to classify the exceptional surgeries along alternatingknots, it suffices to prove that about 30000 links with a condition on the slopes for certain components
do not admit exceptional surgeries. By applying our codes to those links, we see that none of such links
admit any unexpected exceptional surgeries. The computational cost is quite expensive and not durable forstandard personal computers. Hence we used TSUBAME, the supercomputer of Tokyo Tech. Thus with abig aid of computer, we have a classification of the exceptional surgeries along the alternating knots.

備考	 :	 論文要旨は、和文 2000 字と英文 300 語を 1 部ずつ提出するか、もしくは英文 800 語を 1 部提出してください。	 
Note : Thesis Summary should be submitted in either a copy ofWords (English).

2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800