[1/17] The 12th “AIMR Seminar”
Speaker
Dr. Sonia Mahmoudi (SUURI-COOL/Mathematical Science Group, AIMR)
Title
The theory of doubly periodic tangles: a topological study towards applications in materials science
Date
January 17 (Fri), 2025, 14:10–14:55 on Japan Time
Venue
Seminar Room, 2nd floor, AIMR Main Building
Abstract
Doubly periodic tangles (DP tangles) are complex entangled structures consisting of curves embedded in the thickened plane that can be defined as lifts of links in the thickened torus. They serve as a significant framework for analyzing and understanding the topological properties of interwoven filament systems across micro-, meso-, and macro-scales, including, but not limited to, molecular assemblies, polymer melts, fabric-like structures, and cosmic filaments.
Despite the importance of these systems, existing CAD/CAM software is largely limited to visualization of complex entangled structures. Tools for simulating their topology, predicting physical properties, and linking structure to function are still lacking, leaving a significant gap in the design of materials with targeted properties. Addressing this requires a topological study of DP tangles, which provides an important for developing such tools.
The topological classification of DP tangles is at least as hard a problem as the full classification of knots and links in three-space and is approached by constructing topological invariants. To reduce the complexity of the classification problem of DP tangles, a strategy is to consider the quotient of a DP diagram under a periodic lattice, namely a link diagram in the (flat) torus that we call a (flat) motif. This approach leads to a two-dimensional diagrammatical theory of the topological equivalence of DP tangles, which has been established in [1] on the level of motifs, generalizing works initiated by Grishanov et al. related to textiles [2,3].
In this talk, we will classify DP tangles using the well-known Jones polynomial, extended to take into account periodic boundary conditions [4]. Originally developed for classical knots and links, the Jones polynomial is a notable invariant with deep connections to statistical mechanics (via the Potts model) and quantum field theory (via Chern-Simons theory). We will show that the Jones polynomial remains an invariant for DP tangles through the introduction of new topological invariants that we will present [5,6]. These invariants appear as interesting measures to compare the topological complexity of DP tangles. Finally, we will present examples of computations of these invariants on several DP tangles, comparing them with some known numerical invariants.
- [1] I. Diamantis, S. Lambropoulou, S. Mahmoudi, Equivalences of doubly periodic tangles, arXiv:2310.00822 (2023).
- [2] S.A. Grishanov, V.R. Meshkov, A.V. Omel’Chenko, Kauffman-type polynomial invariants for doubly periodic structures, Journal of Knot Theory and its Ramifications, 16, (2007) 779-788.
- [3] H.R. Morton, S.A. Grishanov, Doubly periodic textile structures, Journal of Knot Theory and its Ramifications, 18 (2009) 1597-1622.
- [4] S. Mahmoudi, On the classification of periodic weaves and universal cover of links in thickened surfaces, Communications of the Korean Mathematical Society, 39, No. 4, (2024), 997–1025.
- [5] I. Diamantis, S. Lambropoulou, S. Mahmoudi, Directional invariants of doubly periodic tangles, Symmetry, 16(8), (2024), 968.
- [6] I. Diamantis, S. Lambropoulou, S. Mahmoudi, On the combinatorics of doubly periodic tangles, in preparation.
AIMR Seminars and Tea Time
We hold AIMR Seminars and Friday Tea Time as part of the initiative to promote fusion research.
Seminars are presented by AIMR researchers and visiting scientists who provide a wide range of hot topics from materials science to mathematics. The aims of the AIMR Seminar are to build up mutual understanding, communication, and discussions of each other’s research fields.
Weekly held Friday Tea Time is where AIMR researchers and staff members can relax and talk freely while drinking coffee. Expectations of stimulating conversation may arise among the researchers and leading to new research.
From 2024, active fusion research through synergistic effects from AIMR Seminars and Friday Tea Time will be promoted.
Past AIMR Seminar
| 11th | Jan. 10, 2025 | Prof. Lorenzo Di Bari (University of Pisa) |
|---|---|---|
| 10th | Dec. 13, 2024 | Dr. Yuta Yoshimoto (Fujitsu Ltd.) |
| 9th | Nov. 29, 2024 | Prof. Zhiqiang Mao (Pennsylvania State University, U.S.A.) |
| 8th | July 19, 2024 | Dr. Shuai Wei (Aarhus University, Denmark) |
| 7th | July 5, 2024 | Prof. Henning Schomerus (Lancaster University, U.K.) |
| 6th | May 10, 2024 | Prof. Hidekazu Kurebayashi (University College London, U.K.) |
| 5th | May 1, 2024 | Prof. Michael Hirscher (Max-Planck-Institute for Intelligent Systems, Germany) |
| 4th | Apr. 26, 2024 | Prof. Konstantinos Danas (ELyTMax, CNRS-Tohoku University) |
| 3rd | Mar. 13, 2024 | Prof. Ryoko Tomiyasu (Institute of Mathematics for Industry, Kyushu University) |
| 2nd | Jan. 26, 2024 | Prof. Michael I. Ojovan (Imperial College London, U.K.) |
| 1st | Jan. 12, 2024 | Dr. William Chuck Witt (University of Cambridge, U.K.) |
About Tea Time Talk
As part of an event during Friday Tea Time, AIMR has been hosting lectures by famed researchers and the archives up to 2023 are listed below:
| 43rd | July 21, 2023 | Prof. Feral Temelli (University of Alberta, Canada) |
|---|---|---|
| 42nd | June 30, 2023 | Dr. Douglas Brumley (The University of Melbourne, Australia) |
| 41st | Mar. 28, 2023 | Prof. Motomu Tanaka (Physical Chemistry of Biosystems, Institute of Physical Chemistry, Heidelberg University, Germany) |
| 40th | Nov. 10, 2022 | Prof. Masahiro Yoshimura (National Cheng Kung University, Taiwan) |
| 39th | June 17, 2022 | Prof. Rana Mohtadi (Toyota Research Institute of North America) |
| 38th | June 17, 2022 | Prof. Michael Hirscher (Max-Planck-Institute for Intelligent Systems, Germany) |
| 37th | June 3, 2022 | Prof. Reiko Oda (French National Centre for Scientific Research) |
| 36th | May 28, 2021 | Prof. Magda Titirici (Imperial College London, U.K.) |
| 35th | Mar. 26, 2021 | Prof. Yaroslav Blanter (Kavli Institute of Nanoscience, Delft University of Technology) |
| 34th | Oct. 23, 2020 | Dr. Kazuhito Tsukagoshi (PI:WPI-MANA, NIMS & AIMR) |
| 33rd | July 17, 2020 | Prof. Kaoru Tamada (IMCE, Kyushu University & AIMR) |
| 32nd | Jan. 17, 2020 | Prof. Miho Yamauchi (I2CNER / Kyushu University & AIMR) |
| 31st | Jan. 17, 2020 | Prof. Mickaël Lallart (INSA Lyon, Université de Lyon) |
| 30th | Oct. 25, 2019 | Assoc. Prof. Frédéric Gillot (Department of Solid Mechanics, Mechanical Eng. and Civil Eng., École Centrale de Lyon) |
| 29th | July 12, 2019 | Prof. Qian Niu (Department of Physics, The University of Texas at Austin) |
| 28th | May 17, 2019 | Prof. Damien Fabrègue (INSA Lyon, France / ELyT MaX Lab at Tohoku University) |
| 27th | Dec. 14, 2018 | Prof. Yukitoshi Motome (Department of Applied Physics, The University of Tokyo, Japan) |
| 26th | July 13, 2018 | Dr. Shimpei Ono (Central Research Institute of Electric Power Industry, Japan) |
| 25th | May 11, 2018 | Dr. Pierre-Antoine Geslin (Mateis lab, INSA Lyon/CNRS / ELyTMaX Lab at Tohoku University / Institute for Materials Research, Tohoku University) |
| 24th | Apr. 27, 2018 | Prof. Masaru Tsukada (AIMR, Tohoku University) |
| 23rd | Dec. 8, 2017 | Prof. Jean-Yves Cavaillé (INSA Lyon, France / ELyT MaX Lab at Tohoku University) |
| 22nd | July 28, 2017 | Prof. Nicolas Mary (INSA Lyon, France / ELyT MaX Lab at Tohoku University) |
| 21st | July 21, 2017 | Prof. Gael Sebald (INSA Lyon, France / ELyT MaX at Tohoku University) |
| 20th | July 7, 2017 | Prof. Arun Bansil (Northeastern University) |
| 19th | June 30, 2017 | Dr. Takeshi Nakanishi (AIST-Tohoku U Mathematics for Advanced Materials Open Innovation Laboratory (MathAM-OIL)) |
| 18th | June 23, 2017 | Prof. Denis Arčon (Jožef Stefan Institute, Slovenia) |
| 17th | Apr. 28, 2017 | Prof. Ayumi Hirano (AIMR, Tohoku University) |
| 16th | Nov. 29, 2016 | Dr. David Guy Austing (National Research Council Canada, Canada) |
| 15th | Sept. 30, 2016 | Prof. C. Suryanarayana (University of Central Florida, U.S.A.) |
| 14th | June 17, 2016 | Prof. Tomoteru Fukumura (AIMR, Tohoku University) |
| 13th | May 13, 2016 | Prof. Jean-Yves Cavaillé (ELyT MaX (Joint Lab at Tohoku University), University of Lyon, France) |
| 12th | June 26, 2015 | Dr. B. Muralidharan (Indian Institute of Technology Bombay, India) |
| 11th | June 27, 2014 | Prof. Michael B. Santos (The University of Oklahoma, U.S.A.) |
| 10th | Nov. 19, 2013 | Dr. Pawel Hawrylak (National Research Council Canada, Canada) |
| 9th | Oct. 1, 2013 | Dr. Yuan T. Lee (Nobel laureate in Chemistry, Academia Sinica of Taiwan) |
| 8th | June 28, 2013 | Dr. Bjorn Mysen (Geophysical Laboratory, Carnegie Institution for Science, U.S.A.) |
| 7th | Apr. 12, 2013 | Prof. Yiming Li (National Chiao Tung University, Taiwan) |
| 6th | Jan. 18, 2013 | Prof. Tomasz Dietl (Polish Academy of Sciences, Poland) |
| 5th | Aug. 31, 2012 | Prof. John H. Perepezko (University of Wisconsin-Madison, U.S.A.) |
| 4th | Aug. 17,2012 | Prof. Alain Reza Yavari (Grenoble Institute of Technology, France) |
| 3rd | Aug. 10, 2012 | Prof. Alan Lindsay Greer (University of Cambridge, U.K.) |
| 2nd | July 6, 2012 | Prof. Winfried Teizer (Texas A&M University, U.S.A.) |
| 1st | May 25, 2012 | Prof. Thomas P. Russell (The University of Massachusetts Amherst, U.S.A.) |
Contact
Kazuto Akagi
Deputy Director, Research Support Division, AIMR
| E-mail: | kazuto.akagi.b5@tohoku.ac.jp |
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