arXiv subjects

Toky Andriamanalina

Publications and source records attributed to Toky Andriamanalina.

3 recordsLinked to original sources

arXiv2026

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↗

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↗

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↗