Research on 4-dimensional topology from the viewpoint of graphics and quandle theory

2018 Fiscal Year Annual Research Report

• Project/Area Number: 26287013
• Research Institution: Osaka City University
• Principal Investigator: 鎌田 聖一 大阪市立大学, 大学院理学研究科, 教授 (60254380)
• Co-Investigator(Kenkyū-buntansha): 河内 明夫 大阪市立大学, 大学院理学研究科, 特任教授 (00112524) ; 金信 泰造 大阪市立大学, 大学院理学研究科, 教授 (00152819) ; 大槻 知忠 京都大学, 数理解析研究所, 教授 (50223871) ; 遠藤 久顕 東京工業大学, 理学院, 教授 (20323777) ; 佐藤 進 神戸大学, 理学研究科, 教授 (90345009) ; 安井 弘一 大阪大学, 情報科学研究科, 准教授 (70547009) ; 大城 佳奈子 上智大学, 理工学部, 准教授 (90609091)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: トポロジー / 曲面結び目 / グラフィクス / カンドル / ４次元トポロジー

Outline of Annual Research Achievements

４次元ユークリッド空間に埋め込まれた曲面結び目をmarked graph diagram と呼ばれる平面上に描かれた図式で表示する方法があり、これを用いた曲面結び目の研究が近年盛んになっている。この方法を、４次元ユークリッド空間にはめ込まれた曲面結び目について拡張した。埋め込まれた曲面結び目の marked graph diagram では、すべてのマーカー付き交点においてマーカーに沿った平滑化を施して得られる絡み目図式とマーカーに横断的な方向に平滑化を施して得られる２種類の絡み目図式が、共に自明な絡み目を表すことが条件であった。我々のアイデアは、はめ込まれた曲面結び目の marked graph diagram として、それらの絡み目図式が、H自明と呼ばれる絡み目を表す図式であるという条件に拡張した点である。この結果は、研究代表者と河村建吾によるはめ込まれた曲面結び目の標準形に基づいている。埋め込まれた曲面結び目のmarked graph diagram 表示では、吉川変形と呼ばれる基本変形が知られていたが、はめ込みのケースで、それに追加すべき変形の候補を与えた。
また、H自明な絡み目の成分数が少ないケースについて、リングモーション群というH自明絡み目が４次元空間内を運動する様子を表す群の群表示を与えることに成功した。
平成30年5月30日～6月1日に京都大学数理解析研究所で研究集会「Intelligence of Low-dimensional Topology」を開催した。世話人は大槻知忠（分担者）と伊藤哲也で、11件の講演と約70名（外国人3名を含む）の参加者あった。平成30年9月6日～10日に大阪市立大学で国際研究集会「Four Dimensional Topology」（４次元トポロジー）を開催した。32件の講演と97名（外国人19名を含む）の参加者があった。

Research Progress Status

平成30年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

平成30年度が最終年度であるため、記入しない。

Causes of Carryover

平成30年度が最終年度であるため、記入しない。

Expenditure Plan for Carryover Budget

平成30年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] Presentation of immersed surface-links by marked graph diagrams (2018)
  • Author(s): Seiichi Kamada, Akio Kawauchi, Jieon Kim and Sang Youl Lee
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518500529
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Biquandle cohomology and state-sum invariants of links and surface-links (2018)
  • Author(s): Seiichi Kamada, Akio Kawauchi, Jieon Kim and Sang Youl Lee
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518430162
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Biquandle (co)homology and handlebody-links (2018)
  • Author(s): Atsushi Ishii, Masahide Iwakiri, Seiichi Kamada, Jieon Kim, Shosaku Matsuzaki and Kanako Oshiro
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518430113
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On Gauss codes of virtual doodles (2018)
  • Author(s): Andrew Bartholomew, Roger Fenn, Naoko Kamada and Seiichi Kamada
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518430137
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Doodles on surfaces (2018)
  • Author(s): Andrew Bartholomew, Roger Fenn, Naoko Kamada and Seiichi Kamada
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518500712
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Faithful equivalence of equivalent ribbon surface-links (2018)
  • Author(s): Akio Kawauchi
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518430034
  • Peer Reviewed
• [Journal Article] Classification of a family of ribbon 2-knots with trivial Alexander polynomial (2018)
  • Author(s): Taizo Kanenobu and Toshio Sumi
  • Journal Title: Communications of the Korean Mathematical Society Volume: 33 Pages: 591-604
  • DOI: 10.4134/CKMS.c170222
  • Peer Reviewed
• [Journal Article] On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings (2018)
  • Author(s): T. Ohtsuki and Y. Yokota
  • Journal Title: Math. Proc. Cambridge Philos. Soc. Volume: 165 Pages: 287-339
  • DOI: 10.1017/S0305004117000494
  • Peer Reviewed
• [Journal Article] On the asymptotic expansion of the quantum SU(2) invariant at $q = exp(4 \pi \sqrt{-1}/N)$ for closed hyperbolic 3-manifolds obtained by integral surgery along the figure-eight knot (2018)
  • Author(s): T. Ohtsuki
  • Journal Title: Algebraic & Geometric Topology Volume: 18 Pages: 4187-4274
  • DOI: 10.2140/agt.2018.18.4187
  • Peer Reviewed
• [Journal Article] The pass move is an unknotting operation for welded knots (2018)
  • Author(s): T. Nakamura, Y. Nakanishi, S. Satoh, and A. Yasuhara
  • Journal Title: Topology and its Applications Volume: 247 Pages: 8-19
  • DOI: 10.1016/j.topol.2018.07.005
  • Peer Reviewed
• [Journal Article] Crossing changes, delta moves and sharp moves on welded knots (2018)
  • Author(s): S. Satoh
  • Journal Title: Rocky Mountain Journal of Mathematics Volume: 48 Pages: 967-879
  • Peer Reviewed / Open Access
• [Journal Article] Finiteness of the set of virtual knots with a given state number (2018)
  • Author(s): T. Nakamura, Y. Nakanishi, and S. Satoh
  • Journal Title: Journal of Knot Theory and Its Ramifications Volume: 27 Pages: －
  • DOI: 10.1142/S0218216518500499
  • Peer Reviewed / Open Access
• [Presentation] Graphic descriptions of surface foldings and braided surfaces (2019)
  • Author(s): Seiichi Kamada
  • Organizer: 2019 Winter TAPU Workshop on Knot Theory and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Quandle colorings of knots with cut points (2019)
  • Author(s): Youngju Bae, Naoko Kamada, Seiichi Kamada, Byeorhi Kim
  • Organizer: AMS Spring Central and Western Joint Sectional Meeting
  • Int'l Joint Research / Invited
• [Presentation] Twisted Alexander polynomial of a ribbon 2-knot (2019)
  • Author(s): Taizo Kanenobu
  • Organizer: The 14th East Asian Conference on Geometric Topology
  • Int'l Joint Research / Invited
• [Presentation] Shadow biquandles and local biquandles (2019)
  • Author(s): 大城佳奈子
  • Organizer: 日本数学会2019年度年会
• [Presentation] Calculations of twisted Alexander invariants using $f$-derivatives for quandles (2019)
  • Author(s): Kanako Oshiro
  • Organizer: AMS Spring Central and Western Joint Sectional Meeting
  • Int'l Joint Research
• [Presentation] Two-dimensional braids, their graphical description and application to knot theory (2018)
  • Author(s): Seiichi Kamada
  • Organizer: The 2018 International Conference of the Honam-Youngnam Mathematical Societies
  • Int'l Joint Research / Invited
• [Presentation] On virtual links which are equivalent as twisted links (2018)
  • Author(s): Seiichi Kamada
  • Organizer: The 10th KOOK-TAPU Joint Seminar on Knots and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] On surfaces immersed in 4-space (2018)
  • Author(s): Seiichi Kamada
  • Organizer: Four Dimensional Topology
  • Int'l Joint Research / Invited
• [Presentation] Lifting of surface foldings in dimension 3 to braided surfaces (2018)
  • Author(s): Seiichi Kamada
  • Organizer: The 14th East Asian Conference on Geometric Topology
  • Int'l Joint Research / Invited
• [Presentation] Twisted Alexander polynomial of a ribbon 2-knot of 1-fusion (2018)
  • Author(s): Taizo Kanenobu
  • Organizer: The 10th KOOK-TAPU Joint Seminar on Knots and Related Topics
  • Int'l Joint Research / Invited
• [Funded Workshop] The 14th East Asian Conference on Geometric Topology (2019)
• [Funded Workshop] Four Dimensional Topology (2018)