math.GT daily digest: 10 new submissions for 24 July 2026

New submissions

Writhe-Based Polymer Link Classification Using Machine Learning

• Authors: Jack Beda; Djordje Mihajlovic; Kasturi Barkataki; Davide Michieletto
• arXiv: 2607.20657
• Subjects: Geometric Topology (math.GT); Soft Condensed Matter (cond-mat.soft); Machine Learning (cs.LG)
• Comments: 8 pages, 4 figures

Unique and rapid classification of knots and links is an open mathematical problem that is relevant to a range of (bio)physical systems, including polymer melts, DNA, and proteins. In this paper, we explore a data-driven approach to the classification problem of link topology. Extending the framework introduced in Ref. 1 (Sleiman et al, 2024 Soft Matter, 20(1), pp.71-78), we show that a feedforward neural network trained on the writhe density matrix classifies thermally equilibrated configurations of the first six prime links with 97% accuracy. We demonstrate that this accuracy remains high across a range of temperatures and lengths of link components, while rapidly deteriorating with the addition of topology-altering Gaussian noise; a result consistent with the writhe density matrix containing features sensitive to topology. Our results show that neural networks based on the writhe density matrix efficiently classify two-component links, establishing machine learning as a promising tool for rapid classification of more complex link topologies, e.g. Borromean rings and multi-component links, as the computational cost of exact numerical calculation of topological invariants becomes prohibitive.

On the length sets of closed hyperbolic surfaces

• Authors: Yanlong Hao
• arXiv: 2607.20808
• Subjects: Geometric Topology (math.GT)

For every closed surface of genus , we prove that there exists a countable union of positive-codimension algebraic subsets of Teichmüller space such that, for every hyperbolic metric outside this exceptional set, the number of distinct primitive closed-geodesic lengths at most is bounded below by , where the explicit constant satisfies .

A Geometric Finiteness Theory for Essential Surfaces in Knot Exteriors

• Authors: Makoto Ozawa
• arXiv: 2607.20844
• Subjects: Geometric Topology (math.GT)
• Comments: 102 pages, 21 figures, 2 tables

We develop a relative geometric finiteness theory for essential surfaces in knot exteriors. Let be a unit-thickness representative of a knot type , with , and let be a properly embedded essential surface with and relative thickness at least , defined using positive reach and controlled boundary collars. We prove that every bounded-geometry slice contains only finitely many pair-isotopy classes. We construct explicitly bounded canonical layered codes on a fixed ambient lattice and show that, at resolution with sufficiently fine angular quantization, equality of codes implies ambient pair-isotopy. Thus the topology of each bounded slice is recoverable from finite geometric data. For a fixed exterior, these classes form finite visible subcomplexes of the essential-surface complex; the subcomplexes are monotone, exhaust the full complex, and carry isometric actions levelwise and meridian-preserving actions with controlled reindexing. Positive-reach compactness also yields attainment results for fixed-exterior and compactified visibility problems. Finally, the peripheral geometry gives a writhe window for connected surfaces with nonempty non-meridional boundary: \[ |r|\leq C_{\mathrm{BS}}\Lambda^{4/3}+w(\Delta,\tau). \] This produces slope invisibility gaps and a linear joint-area lower bound for a Seifert surface and cabling annulus of a torus knot. The framework is triangulation-free and complementary to normal-surface, branched-surface, sutured-manifold, and Heegaard-theoretic methods; it does not assert finiteness without geometric bounds.

An Image--Kernel--Reconstruction Program for Essential-Surface Complexes of Knot Exteriors

• Authors: Makoto Ozawa
• arXiv: 2607.20869
• Subjects: Geometric Topology (math.GT)
• Comments: 38 pages, no figures

We study the action of the meridian-preserving mapping class group of a knot exterior E on the disjointness complex of connected two-sided orientable essential surfaces, with natural type, peripheral, homological, and characteristic-submanifold decorations. The underlying complex is Schultens's initial surface complex S_0(E). We organize four rigidity questions: which automorphisms are geometric, which mapping classes are invisible, what characteristic and peripheral structure is intrinsically recoverable, and how much decoration is necessary. A conditional reduction principle separates recognition, global realization, and kernel determination; an intentionally over-marked hierarchy atlas gives a reconstruction benchmark. Classical three-manifold results yield model calculations. For a torus-knot exterior, the complex has two isolated vertices; a type or slope label removes its nongeometric transposition, while the labeled-action kernel is generated by strong inversion. For a connected sum of two nonfibered prime knots, the decomposing annulus is the unique essential annulus, and its twist translates Banks's integer winding coordinate, so the annular-twist subgroup acts faithfully on the Kakimizu complex. For a hyperbolic knot exterior, every kernel considered is finite and has no nontrivial twist subgroup. For the figure-eight knot, the complex has three isolated vertices of slopes 0 and +/-4; the full mapping class group is dihedral of order eight, the geometric image on the four decorated objects considered is Z/2, and the kernel is the orientation-preserving Klein four subgroup. The classifications and symmetry groups are classical; the contribution is the common image-kernel bookkeeping and reconstruction framework.

Binding Numbers of Tight Contact Structures on

• Authors: Csaba Daniel Farkas
• arXiv: 2607.21229
• Subjects: Geometric Topology (math.GT)
• Comments: 19 pages, 10 figures

We study binding numbers of tight contact structures on the lens spaces . Using the invariant, together with restrictions on planar monodromy factorizations and the Durst-Kegel algorithm for computing from open books, we obtain lower bounds for the number of binding components of planar open books supporting these contact structures. As a consequence, we compute the binding number of the universally tight contact structures on , showing that it is equal to .

Planar contact 3-manifolds with infinitely many Stein fillings

• Authors: R. Inanc Baykur
• arXiv: 2607.21303
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 7 pages

We prove that there are infinitely many closed contact 3-manifolds supported by planar open books, each admitting infinitely many pairwise non-homeomorphic Stein fillings. This answers K3 Problem 4.105. As a corollary, there are contact 3-manifolds that admit infinitely many Stein fillings but do not admit arbitrarily large ones.

On the Legendrian invariant in knot lattice homology

• Authors: Sarah Zampa
• arXiv: 2607.21335
• Subjects: Geometric Topology (math.GT)
• Comments: 10 pages, 1 figure, accepted for publication

The Ozsváth-Szabó contact invariant of the link of a normal surface singularity equipped with its canonical contact structure was transposed to lattice homology theory by Bodnár-Plamenevskaya. When considering a transverse algebraic knot in the link, the chain complex computing can be equipped with an Alexander grading, and we can define an element in the bigraded theory , which maps to the contact element by forgetting the filtration. We show that the Alexander grading (as defined by Ozsváth-Stipsicz-Szabó) of this element is invariant under all blow-ups of the underlying plumbing graph. Furthermore, we utilize the fact that for specific types of blow-ups, the resulting lattice chain complexes are filtered chain homotopic and the chains maps map this element in one chain complex to the other, thereby providing a partial combinatorial description of the Legendrian invariant.

Morphing Graphs on Hyperbolic Surfaces

• Authors: Yanwen Luo; Yuan Luo
• arXiv: 2607.21469
• Subjects: Geometric Topology (math.GT); Computational Geometry (cs.CG)
• Comments: 24 pages, 5 figures

We propose the first algorithm to morph geometric graphs on hyperbolic surfaces. It is based on a generalization of Tutte's spring embedding theorem on essentially 3-vertex-connected graphs. We describe the algorithms in detail and show experiments with triangulations and graphs on a hyperbolic surface of genus two, the Bolza surface, and a hyperbolic surface of genus three, the Klein quartic.

Cross submissions

3d-3d correspondence for knot complements with finite and large

• Authors: Hee-Joong Chung
• arXiv: 2607.21479
• Cross-list from: hep-th
• Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)
• Comments: 37 pages

For at finite and large , with a totally symmetric representation, we realize the homological block for a knot complement , given in the form of the inverted Habiro series, as a half-index of a 3d theory by studying some examples, which we expect to extend to general knots. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by taking a certain set of poles. Through the half-index realization, we describe a method for obtaining the homological block and its -deformed version for from a Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary .

Compressed primitivity problem in free groups

• Authors: Ilya Kapovich
• arXiv: 2607.21499
• Cross-list from: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 24 pages, no figures

For a fixed integer , we prove that the \emph{compressed primitivity problem} in the free group is decidable in non-deterministic polynomial time. That is, for a \emph{straight-line program} over representing an element , the problem of deciding whether is primitive in belongs to , with input measured by the size of . For , we prove that this problem is decidable in deterministic polynomial time. We also show that, in every fixed rank , automorphic minimality of the conjugacy class of a compressed word in is decidable in deterministic polynomial time.