arXiv subjects

Sonia Mahmoudi

Publications and source records attributed to Sonia Mahmoudi.

17 recordsLinked to original sources

arXiv2026

The untangling number as a measurement of entanglement complexity in 3-periodic networks

Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the (possibly curvilinear) edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we define the least tangled embeddings of 3-periodic networks that we call ground states, through the use of knot-theoretic crossing diagrams. The concept of a ground state permits the definition of a measure of entanglement complexity called the untangling number that quantifies the distance between a given 3-periodic structure and its least tangled version.

math.GT↗

arXiv2026

Textiles: from twisted yarn to topology and mechanics

While textiles have existed throughout much of human history as complex mechanical metamaterials, textile science has largely been overlooked by the physics community until recently. In this review, we consider the symmetry, topology, and mechanics of woven and knitted materials, showing that they represent a unique, if under-explored, corner of condensed matter. We start with the basic construction and elementary mechanical model of spun yarn, reviewing recent developments twisted bundle structures. We then introduce woven and knitted fabrics as materials with layer symmetries that can be topologically characterized as knots and links in the thickened torus. We finally discuss fabric mechanics and geometry, invoking recent results surrounding yarn-level and stitch-level geometry, deformations, and defect structures.

cond-mat.soft↗

arXiv2026

An Orbifold Framework for Classifying Layer Groups with an Application to Knitted Fabrics

Entangled structures such as textiles and architected materials are often doubly periodic. Due to this property and their finite transverse thickness, the symmetries of these materials are described by the crystallographic layer groups. While orbifold notation provides a compact topological description and classification of the planar wallpaper groups, no analogous framework has been available for the spatial layer groups. In this article we develop an orbifold theory in three dimensions and introduce a complete set of Conway-type symbols for all layer groups. To illustrate its applicability, we analyze several knitted fabric motifs and show how their layer-group symmetries are naturally expressed in this new orbifold notation. This work establishes a foundation for the topological classification of doubly periodic structures beyond the planar setting.

cond-mat.soft↗

arXiv2026

From annular to toroidal knotoids and their universal bracket polynomials

In this paper we study the theory of multi-knotoids in the annulus and in the torus, building up from the theory of planar knotoids to the theory of toroidal knotoids through the theory of annular knotoids. We introduce the concept of lifted annular and toroidal knotoids and examine inclusion relations arising naturally from the topology of the supporting manifolds. We also introduce the concept of mixed knotoids as special cases of planar knotoids, containing a fixed unknot for representing the thickened annulus or a fixed Hopf link for representing the thickened torus. We then extend the Turaev loop bracket for planar knotoids to bracket polynomials for annular and for toroidal knotoids, whose universal analogues recover the Kauffman bracket knotoid skein modules of the thickened annulus and the thickened torus.

math.GT↗

arXiv2026

On the isotopies of tangles in periodic 3-manifolds using finite covers

A periodic tangle is a one-dimensional submanifold in $\mathbb{R}^3$ that has translational symmetry in one, two or three transverse directions. A periodic tangle can be seen as the universal cover of a link in the solid torus, the thickened torus, or the three-torus, respectively. Our goal is to study equivalence relations of such periodic tangles. Since all finite covers of a link lift to the same periodic tangle, it is necessary to prove that isotopies between different finite covers are preserved. In this paper, we show that if two links have isotopic lifts in a common finite cover, then they are isotopic. To do so, we employ techniques from 3-manifold topology to study the complements of such links.

math.GT↗

arXiv2026

Equivalence of Doubly Periodic Tangles

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots and links in the thickened torus, and which can be chosen in infinitely many ways. DP tangles are used in modeling materials and physical systems of entangled filaments. In this paper we establish the complete mathematical framework of the topological theory of DP tangles. We present an exhaustive analysis of DP tangle isotopies. These are distinguished in local isotopies and global isotopies. Our analysis yields the characterization of DP isotopy as an equivalence relation on the level of their (flat) motifs, called DP tangle equivalence. Along the way we also discuss motif minimality. We further generalize our results to other diagrammatic categories, namely framed, virtual, welded, singular, pseudo, tied and bonded DP tangles, which could be used in novel applications.

math.GT↗

arXiv2025

The Theory of Doubly Periodic Pseudo Tangles

Doubly periodic tangles (DP tangles) are configurations of curves embedded in the thickened plane, invariant under translations in two transversal directions. In this paper we extend the classical theory of DP tangles by introducing the theory of {\it doubly periodic pseudo tangles} (pseudo DP tangles), which incorporate undetermined crossings called {\it precrossings}, inspired by the theory of pseudo knots. Pseudo DP tangles are defined as liftings of spatial pseudo links in the thickened torus, called {\it pseudo motifs}, and are analyzed through diagrammatic methods that account for both local and global isotopies. We emphasize on {\it pseudo cover equivalence}, a concept defining equivalence between finite covers of pseudo motif diagrams. We investigate the notion of equivalence for these structures, leading to an analogue of the Reidemeister theorem for pseudo DP tangles. Furthermore, we address the complexities introduced by pseudo cover equivalence in defining minimal pseudo motif diagrams. This work contributes to the broader understanding of periodic entangled structures and can find applications in diverse fields such as textiles, materials science and crystallography due to their periodic nature.

math.GT↗

arXiv2025

From planar to annular to toroidal bracket polynomials for pseudo knots and links

Pseudo links are equivalence classes under Reidemeister-type moves of link diagrams containing crossings with undefined over and under information. In this paper, we extend the Kauffman bracket and Jones-type polynomials from planar pseudo links to annular and toroidal pseudo links and their respective lifts from the three-space to the solid torus and the thickened torus. Moreover, since annular and toroidal pseudo links can be represented as mixed links in the three-sphere, we also introduce the respective Kauffman bracket and Jones-type polynomials for their planar mixed link diagrams. Our work provides new tools for the study of annular and toroidal pseudo links.

math.GT↗

arXiv2025

Topological Defect Propagation to Classify Knitted Fabrics

Knits and crochets are mechanical metamaterials with a long history and can typically be produced from a single yarn. Despite the simplicity of the manufacturing process, they exhibit a wide range of structural configurations with diverse mechanical properties and application potential. Although there has been recent growing interest in textile-based metamaterials, a rigorous topological characterization of what makes a structure knittable has been lacking. In this paper, we introduce a general criterion based on topological constraints that distinguishes knits and crochets from other textile structures. We demonstrate how the introduction of topological defects and their propagation makes this classification practical. Our approach highlights a fundamental link between manufacturing processes and structural fragility. Within this framework, we show how the rationalization of defect propagation unlocks the design of fabrics with controllable resistance to damage.

cond-mat.soft↗

arXiv2025

Diagrammatic representations of 3-periodic entanglements

Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ in its universal cover $\mathbb{R}^3$, where two non-isotopic links in $\mathbb{T}^3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}^3$ through the use of diagrams representing links in $\mathbb{T}^3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}^3$. The final definition of a tridiagram of a link in $\mathbb{T}^3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}^3$ that possess the same TP tangle preimage.

math.GT↗

arXiv2025

The untangling number of 3-periodic tangles

The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry.

math.GT↗

arXiv2024

From annular to toroidal pseudo knots

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.

math.GT↗

arXiv2024

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

Doubly periodic (DP) weaves and polycatenanes are complex entangled structures embedded in the Euclidean thickened plane, invariant under translations in two independent directions. Their topological properties are fully encoded within a quotient space under a periodic lattice, which we refer to as a motif. On the diagrammatic level, a motif is a specific type of link diagram on the torus, consisting of essential closed curves for DP weaves or null-homotopic curves for DP polycatenanes. In this paper, we introduce a combinatorial methodology to construct these motifs from planar DP tilings using the concept of polygonal link transformations. We also present an approach to predict the type of motif that can be constructed from a given DP tiling and a chosen polygonal link method. This approach has potential applications in various disciplines, such as materials science and chemistry.

math.GT↗

arXiv2024

On the classification of periodic weaves and universal cover of links in thickened surfaces

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.

math.GT↗

arXiv2024

Directional Invariants of Doubly Periodic Tangles

In this paper we define novel topological invariants of doubly periodic tangles (DP tangles). DP tangles are embeddings of curves in the thickened plane with translational symmetries in two independent directions. We first organize the components of a DP tangle into different interlinked compounds, which are invariants of a DP tangle. The notion of interlinked compound leads to the classification of DP tangles according to their directional type. We then prove that the directional type is an invariant of DP tangles using the concept of axis-motif, which can be viewed as the blueprint of a DP tangle.

math.GT↗

arXiv2022

Construction and Classification of Combinatorial Weaving Diagrams

This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar tiling with over or under information at each vertex. Therefore, a weaving diagram can be constructed using two sets of cycles, one to build a tiling, and a second to define the crossing information. However, this construction method does not guarantee the uniqueness of the diagram, so we define the notion of equivalence classes of weaving diagrams using the concept of crossing-matrices. Finally, we present a classification of our periodic structures according to the minimum number of crossings on a unit cell.

math.CO↗

arXiv2022

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

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.

math.GT↗