Finite type invariants and Milnor invariants by clasper theory

• Project/Area Number: 16K17586
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hiroshima University (2020); Institute of Physical and Chemical Research (2017-2019); The University of Tokyo (2016)
• Principal Investigator: Kotorii Yuka 広島大学, 先進理工系科学研究科(理), 准教授 (30737143)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 位相幾何学 / 結び目理論 / 絡み目 / ミルナー不変量 / ハンドル体絡み目 / Vassiliev不変量 / Goussarov-Polyak-Viro不変量 / 仮想結び目 / 絡み目ホモトピー / クラスパー理論 / 有限型不変量 / 仮想絡み目 / クラスパー / Milnor不変量

Outline of Final Research Achievements

A link-homotopy is an equivalence relation on links generated by ambient isotopies and self-crossing changes. I constructed link-homotopy invariants for 4-component links and researched properties of link-homotopy classes of links by the invariants. I also researched properties of link-homotopy classes of handle-body links by making invariants. I also researched geometric properties of Goussarov-Polyak-Viro's finite type invariants on virtual links.

Academic Significance and Societal Importance of the Research Achievements

絡み目の不変量をクラスパー理論を用いてクラスパーの言葉で記述することで，図式的な計算が可能となり，計算の簡略化や新しい視点を導入することができる．本研究では，絡み目の絡み目ホモトピー不変量をクラスパーを用いて再定式化を与え，そこからさらに新しい不変量を構成した．このように，クラスパーを用いた定式化は，幾何的な視点からの新しい発展が期待される．

Research Products

• [Journal Article] HL-homotopy of handlebody-links and Milnor's invariants (2017)
  • Author(s): Kotorii Yuka、Mizusawa Atsuhiko
  • Journal Title: Topology and its Applications Volume: 221 Pages: 715-736
  • DOI: 10.1016/j.topol.2017.02.073
  • Peer Reviewed
• [Journal Article] HL-homotopy of handlebody-links and Milnor’s invariants (2017)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Journal Title: Topology and its Applications Volume: 221 Pages: 715-736
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] A relation between Milnor’s μ-invariants and HOMFLYPT polynomials (2016)
  • Author(s): Yuka Kotorii
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 25 Issue: 13 Pages: 1650072-1650072
  • DOI: 10.1142/s0218216516500723
  • Peer Reviewed / Open Access
• [Journal Article] On handlebody-links and Milnor's link-homotopy invariants (2016)
  • Author(s): Yuka Kotorii
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 2004 Pages: 38-46
  • Open Access
• [Journal Article] Relationship between the Milnor's ${\mu}$-invariant and HOMFLYPT polynomial (2016)
  • Author(s): Yuka Kotorii
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1991 Pages: 98-103
  • Open Access
• [Presentation] 絡み目の絡み目ホモトピーによる分類について (2021)
  • Author(s): 小鳥居 祐香，水澤 篤彦
  • Organizer: 広島大学トポロジー・幾何セミナー
  • Invited
• [Presentation] Linking numbers of random links (2020)
  • Author(s): 小鳥居 祐香
  • Organizer: 数学と諸分野の連携にむけた若手数学者交流会2021
  • Invited
• [Presentation] A new presentation of 4-component link-homotopy classes and an algorithm (2020)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: Friday Seminar on Knot Theory 大阪市立大学数学研究所
  • Invited
• [Presentation] clasperを用いた, 2つの4成分linkがlink-homotopicであるか判定するアルゴリズム (2020)
  • Author(s): 小鳥居 祐香，水澤 篤彦
  • Organizer: 日本数学会2020 年度秋季総合分科会, 熊本大学
• [Presentation] Levine's classification of 4-component links up to link-homotopy and its classification by claspers (2020)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: East Asian Conference on Geometric Topology
  • Int'l Joint Research / Invited
• [Presentation] On link-homotopy classes of 4-component links and clasper presentations (2019)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: The 2nd Meeting on Spatial Graph Theory
  • Int'l Joint Research / Invited
• [Presentation] On Milnor's invariants and handle body-links (2019)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: Young Topologist meeting 2019
  • Int'l Joint Research
• [Presentation] Introduction to Milnor invariant -from classical knots to handlebody-links- (2019)
  • Author(s): Yuka Kotorii
  • Organizer: ハンドル体結び目とその周辺 12
  • Invited
• [Presentation] On Levine’s classification of link-homotopy classes of 4-component links (2019)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: Friday Seminar on Knot Theory
  • Invited
• [Presentation] Link-homotopy classes of 4-component links and clasper theory (2019)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: Young Topologist meeting 2019
  • Int'l Joint Research
• [Presentation] クラスパーを用いた4成分絡み目のlink-homotopy 類の分類 (2019)
  • Author(s): 小鳥居 祐香, 水澤 篤彦
  • Organizer: 日本数学会2019年度秋季総合分科会
• [Presentation] On the degree three case of Goussarov-Polyak-Viro conjecture of knot (2019)
  • Author(s): 伊藤昇
  • Organizer: 日本数学会2018年度年会プログラム
• [Presentation] On random knot (2019)
  • Author(s): 小鳥居祐香
  • Organizer: 第二回理研AIP数学系合同セミナー
  • Invited
• [Presentation] 結び目のトポロジー (2019)
  • Author(s): 小鳥居祐香
  • Organizer: 第２回自然科学の数学的原理
  • Invited
• [Presentation] On random models of links and linking numbers (2019)
  • Author(s): 小鳥居祐香
  • Organizer: 京都大学高等研究院平岡研究室セミナー
  • Invited
• [Presentation] On combinatorial formulas of finite type invariant for knots (2018)
  • Author(s): 小鳥居祐香
  • Organizer: 大阪大学トポロジーセミナー
  • Invited
• [Presentation] ミルナー不変量とハンドル体絡み目の分類について (2018)
  • Author(s): 小鳥居祐香
  • Organizer: 第2回岡潔女性数学者セミナー
  • Invited
• [Presentation] Goussarov-Polyak-Viro conjecture of knots with degree three and reduced Polyak algebra (2018)
  • Author(s): 伊藤昇
  • Organizer: 電機大数学講演会
  • Invited
• [Presentation] On a local move related to Goussarov-Polyak-Viro’s finite type invariant on virtual knots (2018)
  • Author(s): 小鳥居祐香
  • Organizer: Tsuda-Gakugei Topology Workshop
  • Invited
• [Presentation] パーシステントホモロジーを用いた位相的データ解析について (2018)
  • Author(s): 小鳥居祐香
  • Organizer: 最適輸送と機械学習理論の周辺,
  • Invited
• [Presentation] Topology and machine leaning (2018)
  • Author(s): 小鳥居祐香
  • Organizer: 理研AIP ニューカマーセミナー
  • Invited
• [Presentation] Goussarov-Polyak-Viro's finite type invariant and a local move on virtual knots (2018)
  • Author(s): Yuka Kotorii
  • Organizer: 10th KOOK-TAPU workshop
  • Int'l Joint Research / Invited
• [Presentation] 結び目のGoussarov-Polyak-Viro予想について (2018)
  • Author(s): 伊藤昇
  • Organizer: 異分野・異業種研究交流会2018
• [Presentation] On Goussarov-Polyak-Viro's finite type invariants and a virtual $C_n$-equivalence on long virtual knots (2018)
  • Author(s): Yuka Kotorii
  • Organizer: 低次元トポロジーセミナー, 京都大学
  • Invited
• [Presentation] $C_n$-move on long virtual knot and Goussarov-Polyak-Viro's (2018)
  • Author(s): 小鳥居 祐香
  • Organizer: 日本数学会2018 年度年会プログラム, 東京大学
• [Presentation] Milnor invariant for bouquet graphs (2017)
  • Author(s): 小鳥居祐香、水澤篤彦
  • Organizer: 第40回伊豆トポロジーセミナー
  • Place of Presentation: ホテル伊東ガーデン (神奈川県・伊東市)
  • Year and Date: 2017-03-20
• [Presentation] Virtual $C_n$-moves on long virtual knots (2017)
  • Author(s): Yuka Kotorii
  • Organizer: Lando-Kazarian Seminar, Higher School of Economics, Russia
  • Invited
• [Presentation] Milnor's link-homotopy invariants for handlebody-links (2017)
  • Author(s): Yuka Kotorii
  • Organizer: Geometric Topology Seminar, Higher School of Economics, Russia
  • Invited
• [Presentation] Virtual $C_n$-moves on long virtual knots (2017)
  • Author(s): Yuka Kotorii
  • Organizer: Manturov seminar, Moscow State University, Russia
  • Invited
• [Presentation] On $Delta$-link homotopy (2017)
  • Author(s): Yuka Kotorii
  • Organizer: Geometric Topology Seminar, Higher School of Economics, Russia
  • Invited
• [Presentation] On virtual $C_n$-moves on virtual knots (2017)
  • Author(s): Yuka Kotorii
  • Organizer: N-KOOK セミナー
  • Invited
• [Presentation] On higher order linking numbers for handlebody-links (2017)
  • Author(s): Yuka Kotorii
  • Organizer: 幾何セミナー, 大阪大学
  • Invited
• [Presentation] 結び目の数学 (2017)
  • Author(s): 小鳥居 祐香
  • Organizer: RIKEN iTHEMS のアウトリーチについての研究会2017
  • Invited
• [Presentation] On link-homotopy classes of handlebody-links (2017)
  • Author(s): Yuka Kotorii
  • Organizer: Friday Seminar on Knot Theory, 大阪市立大学数学研究所
  • Invited
• [Presentation] Milnor invariant and its relationship with Alexander polynomial (2016)
  • Author(s): 小鳥居祐香
  • Organizer: 筑波大学トポ ロジーセミナー
  • Place of Presentation: 筑波大学 (茨城県・つくば市)
  • Year and Date: 2016-11-16
  • Invited
• [Presentation] Milnor invariant for handlebody-links (2016)
  • Author(s): 小鳥居祐香、水澤篤彦
  • Organizer: 2016 日本数学会秋季総合分科会
  • Place of Presentation: 関西大学(大阪府・吹田市)
  • Year and Date: 2016-09-15
• [Presentation] On Milnor’s invariants for handlebody-links (2016)
  • Author(s): 小鳥居祐香、水澤篤彦
  • Organizer: 茨城高専数学セミナー
  • Place of Presentation: 茨城工業高等専門学校 (茨城県・ひたちなか市)
  • Year and Date: 2016-09-02
  • Invited
• [Presentation] On Milnor’s $\bar{\mu}$-invariants for handlebody-links (2016)
  • Author(s): Yuka Kotorii, Mizusawa Atsuhiko
  • Organizer: International Workshop on Spatial Graphs 2016
  • Place of Presentation: 早稲田大学 (東京都・新宿区)
  • Year and Date: 2016-08-03
  • Int'l Joint Research
• [Presentation] 結び目の不変量 (2016)
  • Author(s): 小鳥居祐香
  • Organizer: 無限群と幾何学の新展開のアウトリーチについての研 究会
  • Place of Presentation: 玉原セミナーハウス (群馬県・沼田市)
  • Year and Date: 2016-06-10
  • Invited
• [Presentation] On handlebody-links and Milnor’s link-homotopy invari- ants (2016)
  • Author(s): Yuka Kotorii, Atsuhiko Mizusawa
  • Organizer: Intelligence of Low-dimensional Topology
  • Place of Presentation: RIMS (京都府・京都市)
  • Year and Date: 2016-05-18
  • Int'l Joint Research / Invited
• [Presentation] On Milnor’s link-homotopy invariants for handlebody-links (2016)
  • Author(s): 小鳥居祐香、水澤篤彦
  • Organizer: トポロジー火曜セミナー
  • Place of Presentation: 東京大学 (東京都・目黒区)
  • Year and Date: 2016-05-10
  • Invited
• [Presentation] ハンドル体絡み目のミルナー不変量と HL-ホモトピー分類について (2016)
  • Author(s): 小鳥居祐香、水澤篤彦
  • Organizer: 東京女子大学トポロジーセミナー
  • Place of Presentation: 東京女子大学 (東京都・杉並区)
  • Year and Date: 2016-04-23
  • Invited