Details of the Researcher

PHOTO

Mahmoudi Ep Sato Sonia
Section
Advanced Institute for Materials Research
Job title
Assistant Professor
Degree

  • 博士（理学）（東北大学）

  • Ｍ．Ｓ．（IMT Mines Ales）

Research History 5

  • 2023/11 - Present
    Tohoku University　WPI-AIMR (SUURI-COOL)　Assistant Professor

  • 2022/12 - 2023/10
    Tohoku University　WPI-AIMR　Visiting Researcher (Kotani Lab)

  • 2022/12 - 2023/10
    Drexel University and University of Pennsylvania　Center of Functional Fabrics　Research Assistant Professor

  • 2022/10 - 2022/11
    Tohoku University　WPI-AIMR (Kotani Lab)　Postdoc (JSPS PD)

  • 2019/10 - 2022/09
    Tohoku University　WPI-AIMr (Kotani Lab)　Research Assistant

Education 1

  • Tohoku University　Graduate School of Science　Department of Mathematics

    2019/10 - 2022/09

Research Interests 2

  • Periodic Tangles

  • Knot Theory

Awards 1

  1. Tohoku University 2022 President's Award

    2023/03　Tohoku University

Papers 7

  1. Diagrammatic representations of 3-periodic entanglements Peer-reviewed

    Toky Andriamanalina, Myfanwy E. Evans, Sonia Mahmoudi

    Topology and its Applications　368　109346-109346　2025/07

    Publisher: Elsevier BV

    DOI： 10.1016/j.topol.2025.109346 　

    ISSN： 0166-8641

    More details Close

    Diagrams enable the use of various algebraic and geometric tools in analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures, which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. These diagrams require an extended set of new moves in addition to the Reidemeister moves, which we show to preserve ambient isotopies of triply periodic entangled structures. We use the diagrams to define the crossing number and the unknotting number of the triply periodic entanglements, demonstrating the practicality of the diagrammatic representation.

  2. On the classification of periodic weaves and universal cover of links in thickened surfaces Peer-reviewed

    Sonia Mahmoudi

    Communications of the Korean Mathematical Society　39　(4)　997-1025　2024/10/31

    Publisher: arXiv

    DOI： 10.4134/CKMS.C230171 　

    More details Close

    A periodic weave is the lift of a particular link embedded in a thickened surface to the universal cover. Its components are infinite unknotted simple open curves that can be partitioned in at least two distinct sets of threads. The classification of periodic weaves can be reduced to the one of their generating cells, namely their weaving motifs. However, this classification cannot be achieved through the classical theory of links in thickened surfaces since periodicity in the universal cover is not encoded. In this paper, we first introduce the notion of hyperbolic periodic weaves, which generalizes our doubly periodic weaves embedded in the Euclidean thickened plane. Then, Tait First and Second Conjectures are extended to minimal reduced alternating weaving motifs and proved using a generalized Kauffman bracket polynomial defined for periodic weaving diagrams of the Euclidean plane and generalized to the hyperbolic plane. The first conjecture states that any minimal alternating reduced weaving motif has the minimum possible number of crossings, while the second one formulates that two such oriented weaving motifs have the same writhe.

  3. From Annular to Toroidal Pseudo Knots

    Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi

    Symmetry　2024/10/13

    Publisher: arXiv

    DOI： 10.3390/sym16101360 　

    More details Close

    In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar ${\rm O}$-mixed and ${\rm H}$-mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of ${\rm O}$-mixed and ${\rm H}$-mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus.

  4. Directional Invariants of Doubly Periodic Tangles

    Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi

    Symmetry　2024/07/30

    DOI： 10.3390/sym16080968 　

  5. Equivalence of Doubly Periodic Tangles

    Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi

    2023/10

    Publisher: arXiv

    DOI： 10.48550/ARXIV.2310.00822 　

    More details Close

    Doubly periodic tangles, or \textit{DP tangles}, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are completely defined by their generating cells, the {\it flat motifs}, which can be chosen in infinitely many ways. DP tangles are used in modelling materials and physical systems of entangled filaments. In this paper we establish the mathematical framework of the topological theory of DP tangles. We first introduce a formal definition of DP tangles as topological objects and proceed with an exhaustive analysis in order to characterize the notion of {\it equivalence} between DP tangles and between their flat motifs. We further generalize our results to other diagrammatic categories, such as framed, virtual, singular, pseudo and bonded DP tangles, which could be used in novel applications.

  6. Construction of weaving and polycatenane motifs from periodic tilings of the plane

    Mizuki Fukuda, Motoko Kotani, Sonia Mahmoudi

    2023/04/05

    Publisher: arXiv

    DOI： 10.48550/ARXIV.2206.12168 　

    More details Close

    Doubly periodic weaves and polycatenanes embedded in the thickened Euclidean plane are three-dimensional complex entangled structures whose topological properties can be encoded in any generating cell of its infinite planar representation. Such a periodic cell, called motif, is a specific type of link diagram embedded on a torus consisting of essential simple closed curves for weaves, or null-homotopic for polycatenanes. In this paper, we introduce a methodology to construct such motifs using the concept of polygonal link transformations. This approach generalizes to the Euclidean plane existing methods to construct polyhedral links in the three-dimensional space. Then, we will state our main result which allows one to predict the type of motif that can be built from a given planar periodic tiling and a chosen polygonal link method.

  7. Classification of doubly periodic untwisted (p,q)-weaves by their crossing number and matrices

    Mizuki Fukuda, Motoko Kotani, Sonia Mahmoudi

    Journal of Knot Theory and Its Ramifications　2023/04/05

    DOI： 10.1142/S0218216523500323 　

    More details Close

    A weave is the lift to the Euclidean thickened plane of a set of infinitely many planar crossed geodesics, that can be characterized by a number of sets of threads describing the organization of the non-intersecting curves, together with a set of crossing sequences representing the entanglements. In this paper, the classification of a specific class of doubly periodic weaves, called untwisted (p,q)-weaves, is done by their crossing number, which is the minimum number of crossings that can possibly be found in a unit cell of its infinite weaving diagrams. Such a diagram can be considered as a particular type of quadrivalent periodic planar graph with an over or under information at each vertex, whose unit cell corresponds to a link diagram in a thickened torus. Moreover, considering that a weave is not uniquely defined by its sets of threads and its crossing sequences, we also specify the notion of equivalence classes by introducing a new parameter, called crossing matrix.

Show all ︎Show first 5

Presentations 22

  1. Constructing Periodic Tangles from Tilings Invited

    Sonia Mahmoudi

    2025 Spring Symposium, WPI-SKCM2, Hiroshima University　2025/03

  2. On Doubly Periodic (DP) Tangles & Pseudo DP Tangles Invited

    Sonia Mahmoudi

    2024 Winter School, WPI-SKCM2, Hiroshima University　2024/12

  3. A New Topological Model of Knitting

    Sonia Mahmoudi

    Knitting Day – iTHEMS Math Workshop　2024/11

  4. Topological Modeling of Textiles Towards a Sustainable Industry

    Sonia Mahmoudi

    Workshop: iTHEMS Science Outreach Workshop 2024, Tohoku University　2024/11

  5. On the bracket polynomial of periodic tangles

    Sonia Mahmoudi

    Tohoku Knot Seminar　2024/10

  6. Embedding of weaving motifs in the thickened torus and their periodic covers in the 3-space Invited

    Sonia Mahmoudi

    The 4th International Conference on Surfaces, Analysis, and Numerics in Differential Geometry and the 1st IMAG-OCAMI Joint Conference on Differential Geometry　2024/02

  7. Exploring Periodic Entangled Structures in Materials Science Through Knot Theory Invited

    Sonia Mahmoudi

    Kyoto Winter School 2024 “Towards Holistic Understanding of Life”, Kyoto University　2024/02

  8. Knot Theory in Doubly Periodic Tangles and Applications Invited

    Sonia Mahmoudi

    iTHEMS Math Seminar　2024/01

  9. A Topological Model of Textile Structures

    Sonia Mahmoudi

    ICIAM 2023 Tokyo　2023/08

  10. Topological Model of Weaves and Links in the Thickened Torus Invited

    Sonia Mahmoudi

    Drexel University Math Seminar　2023/06

  11. Equivalence Classes of Doubly Periodic (p,q)-Weaves Invited

    Sonia Mahmoudi

    University of Pennsylvania Math Seminar　2023/04

  12. Doubly Periodic Weaves & Polycatenanes Invited

    Sonia Mahmoudi

    Topology & Computer 2022　2022/10

  13. Doubly Periodic Weaves & Polycatenanes

    Sonia Mahmoudi

    2022 Annual Meeting of the Japan Society of Applied Mathematics　2022/09

  14. A Topological Model of Weavings Invited

    Sonia Mahmoudi

    The Interdisciplinary World of Tangling, University of Potsdam　2022/09

  15. Weaving invariants

    Sonia Mahmoudi

    Hokuriku Knot Seminar 2022　2022/09

  16. Doubly Periodic Entangled Motifs from Planar Tilings

    Sonia Mahmoudi

    Fico González-Acuña Low Dimensional Topology Seminar　2022/08

  17. Periodic Weaving Diagrams

    Sonia Mahmoudi

    CIRM Workshop: Structures on Surfaces　2022/05

  18. An Introduction to the Topology of Weaving Invited

    Sonia Mahmoudi

    Lounge Seminar, School of Applied Mathematics NTUA Athens　2022/05

  19. Classification of Combinatorial Weaving Diagrams

    Sonia Mahmoudi

    18th Mathematics Conference for Young Researchers, March　2022/03

  20. A Topological Model of Weaves

    Sonia Mahmoudi

    CREST Seminar　2022/03

  21. Construction of Weaving Diagrams from Tilings

    Sonia Mahmoudi

    18th Joint Presentation of the Japan Society for Industrial and Applied Mathematics,　2022/03

  22. Equivalence classes of doubly periodic untwisted (p,q)-weaves,

    Sonia Mahmoudi

    5th Mathematical Freshman Seminar　2022/02

Show all Show first 5

Research Projects 1

  1. Construction and Classification of Weaves

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for JSPS Fellows

    Institution: Tohoku University

    2022/04/22 - 2024/03/31