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

絡み目の量子不変量とフルツイスト

Title(English)

Quantum link invariants and full twists

著者(和文)

中兼啓太

Author(English)

Keita Nakagane

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11865号,
授与年月日:2021年3月26日,
学位の種別:課程博士,
審査員:KALMAN TAMAS,遠藤 久顕,本多 宣博,五味 清紀,野坂 武史

Citation(English)

Degree:Doctor (Science),Conferring organization: Tokyo Institute of Technology,
Report number:甲第11865号,
Conferred date:2021/3/26,
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

Powered by T2R2 (Tokyo Institute Research Repository)

（博士課程）Doctoral Program論 文 要 旨ＴＨＥＳＩＳ ＳＵＭＭＡＲＹ

系・コース：
Department of, Graduate major in

数学
数学

学生氏名：
Student’s Name

中兼 啓太系コース

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

指導教員（主）：Academic Supervisor(main)（理学）

Tamás Kálmán

指導教員（副）：Academic Supervisor(sub)要旨（英文 800 語程度）
Thesis Summary （approx.800 English Words ）The HOMFLY polynomial is an invariant of oriented links in the three-dimensional real coordinate
space, which can be computed from link diagrams by using a recursive formula called skein relation.
Khovanov and Rozansky introduced its categorification, called HOMFLY homology. That is to say, acertain graded Euler characteristic of the homology is the HOMFLY polynomial. Although theseinvariants have been well studied since their discoveries, their essential nature is still quite unclear.
The HOMFLY homology has been studied mainly from the viewpoint of algebraic geometry,
representation theory, and mathematical physics. However, it is hardly studied from the viewpoint ofpure topology. One obstacle here is that all known approaches to the homology rely on braidpresentations. In other words, we can compute the HOMFLY homology of a link only fromcombinatorial data of braid presentations of the link. Although the definitions of chain complexes by
Khovanov and Rozansky or by Rasmussen can be naturally applied to every link diagram, theirhomologies do not coincide with the HOMFLY homology of the link. In this thesis, we try to givesome hints to overcome this obstacle by proving some properties of the HOMFLY polynomial andhomology.
We first discuss the full twist formula for the HOMFLY polynomial by Kálmán. The formula is aphenomenon on the lowest and highest v-degree part of the polynomial in terms ofMorton—Franks—Williams bounds. If we apply a positive full twist on a braid presentation of a link,then the coefficients of the highest v-degree part in the HOMFLY polynomial of the link obtained after
full twisting does coincide with the coefficients of the lowest v-degree part in the HOMFLY
polynomial of the original link. We will show that the full twist formula is indeed a common
topological property between the HOMFLY polynomial and the HOMFLY homology. The other topics
around the HOMFLY homology are also discussed in the thesis, and it consists of 6 chapters roughlysummarized below.
In chapter 1, we overview the background of this field of study. We also explain some remaining
problems in the field, such as Dunfield—Gukov—Rasmussen conjecture.
In chapter 2, we review the definitions of all needed concepts, starting from links and braids andending with the HOMFLY polynomial and the HOMFLY homology. After that, we discuss theirknown properties and computational techniques which are used in the rest. We explain the full twist

formula for the HOMFLY polynomial in detail here, which is one of the main targets of this thesis.
In chapter 3, using Mellit’s result, we show the full twist formula for the HOMFLY homology in thecase of positive toric braids. Our proof uses a combinatorial argument over Dyck paths, which shows
an interesting correspondence between focused parts of the HOMFLY homologies.
In chapter 4, we generalize the result in chapter 3 (the full twist formula for the HOMFLY homology)
to all braids, by categorifying the original proof by Kálmán. More precisely, we use Khovanov’s
definition of the HOMFLY homology, which is known as a categorification of the Jones—Ocneanu
trace on the (type A) Hecke algebra. We introduce partial traces on the Hecke algebra and theircategorification, and we show the full twist formula as its application. As a corollary, we see that thefull twist acts as a kind of Serre functor on the homotopy category of Soergel bimodules. This chapter
is based on a joint work with Eugene Gorsky, Matthew Hogancamp, and Anton Mellit.
In chapter 5, we introduce a certain class of link diagrams, called knitted diagrams, which includes all
closed braid diagrams. A knitted diagram is obtained by arranging several braid diagrams on the planeand connecting them appropriately. We show a generalized version of the full twist formula for theHOMFLY polynomial in the class, by using the refined Morton—Franks—Williams bounds by
Murasugi and Przytycki. We hope that this result suggests a hint for finding a way to compute theHOMFLY homology from the combinatorial information of general link diagrams. Moreover, an open
question is given here; can we show a homology version of the full twist formula for knitted
diagrams?
In chapter 6, we compute the lowest q-degree part of the graded Euler characteristic of the homology
for general link diagrams, by using Rasmussen’s definition. If it was a link invariant, then the result
would give the signed interior polynomial of the Seifert graph, but in fact, it gives a specialization of
the Tutte polynomial of the graph. This result leads us to a future problem to categorify the interior
polynomial.

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 of 2000 Japanese Characters and 300 Words (Engｌish) or 1copy of 800
Words (English).
注意：論文要旨は、東工大リサーチリポジトリ（T2R2）にてインターネット公表されますので、公表可能な範囲の内容で作成してください。
Attention: Thesis Summary will be published on Tokyo Tech Research Repository Website (T2R2).