math.GT daily digest: 10 submissions for 28 September 2026

New submissions

Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation

• Authors: Atsuhide Mori
• arXiv: 2609.30406
• Comments: 18 pages. Generative AI (ChatGPT, Claude) used for literature search and verification, checking elementary computations, and mathematical discussion; see the declaration before the references
• Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Symplectic Geometry (math.SG); Quantum Physics (quant-ph)

We study planar open books obtained by lifting braids through branched covers of the disk D^2, together with the quantum operations in the Ising representation. We note a criterion for the associated Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, every CY factorization has minimal length, and its filling minimizes \chi and b_2. An application to Baykur's recent examples gives one planar contact 3-manifold with infinitely many non-homeomorphic CY fillings. The cover there is of degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and the filling is unique in a certain case; at degree \le 3 the filling is unique and is CY under a mild condition. Admissible cuts of the covering disk decompose the openbook into subopenbooks. Every positive factorization then localizes to the pieces, and the CY condition holds exactly when it holds locally. This makes the state space a direct sum of tensor products indexed by the compatible parity choices with at most two qubits in each factor when the pieces have degree \le 4, which is also the range in which the CY condition depends only on the monodromy. For a particular degree 4 cover, the points, lines and flags of the two-qubit doily are realized by the system of its subopenbooks. Fifteen liftable braids there share the same quantum operation and the same Stein filling, and are separated only by which subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift and not by the braid group representation, which suggests a link between contact topology and quantum entanglement.

A Concordance Invariant For Knots from

• Authors: Kashti Satish Umare
• arXiv: 2609.30425
• Comments: 15 pages, 6 figures
• Subjects: Geometric Topology (math.GT)

We construct a multi-filtered smooth knot concordance invariant, , derived from the knot Floer complex . $\beta$ is a three-variable piecewise linear function defined on . We demonstrate that $\beta$ satisfies a subadditivity relation. We apply $\beta$ to prove concordance results for a certain family of L-space knots. We show that the positive cones spanned by Teragaito's knots and are disjoint in the smooth concordance group $\mathcal{C}$. For positive L-space knots, we show $\beta$ completely determines the Alexander polynomial and distinguishes knots with identical ${\rm Y}$ invariants.

Short homology bases for translation surfaces

• Authors: Peter Buser; Achintya Dey; Eran Makover; Bjoern Muetzel
• arXiv: 2609.30561
• Comments: 14 page, 4 figures
• Subjects: Geometric Topology (math.GT)

Let $S$ be a closed translation surface of genus with . We show that for any there exist homologically independent simple closed curves of length at most $ C(\lambda) \cdot \log(g)$, where $C(\lambda )$ is a constant that depends only on $\lambda$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.

Topological components of surface group representations into the unitary group

• Authors: Xueyuan Wan
• arXiv: 2609.30603
• Subjects: Geometric Topology (math.GT)

We study representations of compact oriented surface groups into $\mathrm{U}(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component is path-connected. For genus and boundary components, there are components; for and , there are . The disk case is empty, while the closed-surface representation spaces are connected. The same component classification holds after taking the quotient by conjugation. Our proofs use the boundary rho invariant and explicit matrix deformations. For a three-holed sphere, the representation components have the homotopy types of complex Grassmannians, and their conjugation quotients are contractible.

Hexagon decompositions and the Weil-Petersson metric

• Authors: Marie Abadie
• arXiv: 2609.31278
• Comments: 75 pages, 10 figures
• Subjects: Geometric Topology (math.GT)

We give an explicit quasi-isometry from the flip graph of triangulations to the $ϵ$-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry $Q$ from the hexagon graph to the augmented Teichmüller space equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between and . By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most . We also provide an explicit projection from any point in to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.

Horizon saddle connections and Veech groups of infinite-type dilation surfaces

• Authors: Oscar Rutilio Molina Medrano
• arXiv: 2609.31447
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $ S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R}) $ can be realized as the Veech group of a dilation surface homeomorphic to $ S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.

On the Relation Between Non-Semisimple TQFTs

• Authors: Marco De Renzi
• arXiv: 2609.31449
• Comments: 77 pages
• Subjects: Geometric Topology (math.GT); Category Theory (math.CT); Quantum Algebra (math.QA)

The goal of this paper is to clarify the relation between the Kerler-Lyubashenko TQFT associated with the adjoint end $\mathcal{E}$ of a (not necessarily semisimple) modular category $\mathcal{C}$ and its renormalized version based on the modified trace supported by the ideal $\mathrm{Proj}(\mathcal{C})$ of projective objects of $\mathcal{C}$. More precisely, we construct a $3$-dimensional ETQFT that contains both and . The ETQFT is given by a $2$-functor whose source is the $2$-category of admissible $3$-dimensional cobordisms, and whose target is the $2$-category of finitely complete linear categories. We improve on earlier versions of the construction by dropping the admissibility condition for surfaces. By doing so, we obtain an ETQFT whose circle category is equivalent to the category $\mathcal{C}$, as opposed to the ideal $\mathrm{Proj}(\mathcal{C})$.

Profinite Non-Rigidity of Arithmetic Lattices and the Kähler Property

• Authors: Yukun Du; Feng Hao; Kejia Zhu
• arXiv: 2609.31523
• Comments: 22 pages, comments welcome
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Group Theory (math.GR)

We study the profinite non-rigidity of arithmetic lattices and its implications for the Kähler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the Kählerness of finitely presented, residually finite groups is not determined by its profinite completion.

On Line-Hyperplane Structures for Hitchin Representations

• Authors: Parker Evans; Andrea Tamburelli
• arXiv: 2609.31567
• Comments: 65 pages, 4 figures, 2 tables
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let and be a Hitchin representation. We study the topology of a cocompact domain of discontinuity in the flag manifold of line-hyperplane pairs in defined by Guichard-Wienhard. In particular, we lift $\Omega$ to its -cover in the Stiefel manifold . The domain is highly connected and smoothly fibers over hyperbolic space with unknown fiber , a closed -manifold. We determine the homeomorphism type of as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.

Cross submissions

KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery

• Authors: Hakan Akgün; Xianquan Yan; Kehan Liu; Zhaoyun Chen; Ching Hua Lee
• arXiv: 2609.31152
• Cross-list: cs.MS
• Comments: 40 pages, 15 figures, 4 tables; includes Supplementary Information. Source code, documentation, workflows and data are available at this https URL
• Subjects: Mathematical Software (cs.MS); Computational Geometry (cs.CG); Symbolic Computation (cs.SC); Geometric Topology (math.GT); Computational Physics (physics.comp-ph)

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.