math.GT daily digest: 12 submissions for 2 November 2026

New submissions

Systolic Geometry and Combinatorics of Surfaces

• Authors: Achintya Dey
• arXiv: 2610.40750
• Comments: Ph.D. thesis, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, 2026
• Subjects: Geometric Topology (math.GT)

This thesis investigates two themes. The first studies systolic embeddings of graph on translation surfaces, proving existence results for essential and cellular systolic embeddings and determining extremal genera for certain graph families. The second examines hyperbolic surfaces in the Thurston set, constructing a one-parameter deformation that increases the sys- tole length and produces Bolza-like surfaces, i.e., surfaces whose systolic geodesics decompose the surfaces into triangles, for infinitely many genera. A symmetry property of geodesic inter- sections on these surfaces is also established, together with an application of these Bolza-like surfaces.

Exotic involutions on irreducible 4-manifolds from Real Seiberg--Witten theory

• Authors: Woohyeok Jo, Simone Tagliente
• arXiv: 2610.40802
• Comments: 24 pages, no figures, comments are welcome
• Subjects: Geometric Topology (math.GT)

Using Real Seiberg--Witten theory, Miyazawa constructed infinitely many exotic involutions on , and Baraglia extended this construction to for a broad class of $4$-manifolds $X$ equipped with an involution $c$. We show that, under suitable Real Seiberg--Witten hypotheses, the summand can be omitted when $X$ is non-spin and the complement of the fixed locus is simply-connected. Our construction also yields infinite families of topologically isotopic but smoothly non-isotopic embeddings of the branch surface in the quotient $X/⟨c⟩$. As applications, we construct anti-holomorphic involutions on irreducible, simply-connected projective surfaces of general type with infinitely many exotic copies, none of which is anti-symplectic with respect to any orientation-compatible symplectic form. These surfaces include cyclic branched covers of and complete intersections of multidegree $(2,4k)$ in .

On lens spaces which may embed in the complex projective plane

• Authors: Giulia Carfora
• arXiv: 2610.40957
• Comments: Comments are welcome!
• Subjects: Geometric Topology (math.GT)

Using an obstruction derived by Owens and relying on Donaldson's Diagonalisation Theorem, we obtain a complete classification of lens spaces of Lisca type $(2)$ or $(3)$ that are unobstructed from embedding in a homology complex projective plane. This classification is further refined for lens spaces with $p$ even, using an obstruction due to Lidman--Moore--Vazquez based on work of Lin. This article is extracted from the author's PhD thesis.

Handlebodies separate elements of surface groups

• Authors: Richard Hain
• arXiv: 2610.40987
• Comments: 4 pages
• Subjects: Geometric Topology (math.GT)

In this note I prove that every non-trivial element of the fundamental group of a closed orientable surface X is non-trivial in the fundamental group of some handlebody with boundary X. This result appears to be folklore. I have uploaded a proof as the result is needed in a joint project with Clement Dupont on mapping class group actions on configuration spaces.

Lengths of closed geodesics on random non-compact hyperbolic surfaces

• Authors: Malo Gatignol, Joe Thomas
• arXiv: 2610.41002
• Subjects: Geometric Topology (math.GT); Probability (math.PR); Spectral Theory (math.SP)

We study the asymptotic distribution of the number of primitive, closed geodesics on Weil-Petersson random hyperbolic surfaces with genus $g$ and $n(g)$ cusps, under various growth rates of $n(g)$ as . We prove that the limiting distribution is Poisson with parameter depending explicitly on the limit of . This work is inspired by that of Mirzakhani and Petri's on closed surfaces and Hide and Thomas when $g$ is fixed. Our proof relies on first order expansions for Weil-Petersson moduli space volumes in the regime. We also obtain some new existence results for small Laplacian eigenvalues.

The Table Problem for Convex Domains

• Authors: Ali Naseri Sadr
• arXiv: 2610.41010
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We prove a generalized Table Theorem for bounded convex domains in the plane. Then we prove a variation of the Table Problem for compactly supported functions on the plane. Our method is based on relating the table problem on a surface to the square peg problem for a configuration of star-shaped curves in the tangent bundle of the surface.

Rationally slice knots that are not biprojectively H-slice

• Authors: Elise Brod
• arXiv: 2610.41996
• Comments: 48 pages, 17 figures
• Subjects: Geometric Topology (math.GT)

We show that rationally slice knots need not be biprojectively H-slice (BPH-slice), answering a question in the study of the rational knot concordance group. Our main tool is a lattice embedding obstruction to H-sliceness in for knots with vanishing signature, extending work of Kjuchukova--Miller--Ray--Sakallı. We apply the obstruction to an infinite family of rationally slice knots, beginning with . As additional applications, we obstruct the BPH-sliceness of a family of 2-bridge knots which is not obstructed by the instanton knot concordance invariant , we obstruct a non-alternating family of Montesinos knots, and we give prime alternating knots that are topologically but not smoothly biprojectively H-slice. We discuss why these knots remain plausible candidates for detecting exotic smooth structures on and suggest natural sources of further candidates.

Constructing Lagrangian surfaces in all symplectic toric $4$-manifolds from multi-grid diagrams

• Authors: Shintaro Fushida-Hardy, Laura Wakelin
• arXiv: 2610.42076
• Comments: 50 pages, 40 figures
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We present a new diagrammatic construction of Lagrangian surfaces in all symplectic toric $4$-manifolds from objects called multi-grid diagrams. These are a generalisation of triple grid diagrams for Lagrangian surfaces in wherein the underlying trisection is replaced by the more general notion of a toric decomposition. We give examples of multi-grid diagrams for Lagrangian surfaces in a range of symplectic toric $4$-manifolds, including Evans' recent non-squeezable Klein bottle in .

Polynomial-Time Algorithms for Links

• Authors: Thiago de Paiva, Yi Liu
• arXiv: 2610.42077
• Comments: 30 pages
• Subjects: Geometric Topology (math.GT)

Every link in the $3$-sphere admits a generalised $T$-link presentation. In this paper, we give an explicit polynomial-time algorithm which, given a signed Artin braid word and the number of strands, computes such a presentation. Combining this construction with Yamada--Vogel braiding gives a polynomial-time conversion from arbitrary oriented combinatorial link diagrams to generalised $T$-link parameters. We prove polynomial bounds on the number of strands and crossings in the resulting braid.

As an application, we prove that a link property or a link invariant is computable in polynomial time from diagrams if and only if the corresponding problem is solvable in polynomial time from generalised $T$-link parameters, with parameter input measured by the total number of strands and crossings in the corresponding braid diagram. Whenever a classification criterion or an explicit invariant formula is available in these parameters, it can be applied to the original link by substituting the parameters obtained from the diagram. A polynomial-time test or evaluation then gives a polynomial-time procedure on diagrams. We also establish the corresponding transfer for approximating real-valued invariants to prescribed accuracy.

Holonomies of hyperbolic surfaces with one branch point

• Authors: Bertrand Deroin, Nicolas Tholozan
• arXiv: 2610.42079
• Comments: 52 pages, 14 figures
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let $S$ be a closed oriented surface of genus . We prove that any representation of into $\mathrm{PSL}(2,\mathbb{R})$ with Euler class is the holonomy of a hyperbolic metric on $S$ with one simple branch point.

Cross submissions

Orientations for skein-valued curve counting

• Authors: Brett Parker
• arXiv: 2610.41233
• Cross-list: math.SG
• Comments: 73 pages, 25 figures
• Subjects: Symplectic Geometry (math.SG); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

Ekholm and Shende's skeins-on-branes formalism provides invariant counts of holomorphic curves with boundary on a Maslov zero Lagrangian within a Calabi-Yau 3-fold. This paper proves the orientation relations required for worldsheet skein relations to cancel codimension-one boundary strata. However, standard Cauchy-Riemann Fredholm orientations reveal an unexpected sign puzzle: the orientation relation at a curve with a hyperbolic boundary crossing depends on whether the crossing lies on distinct boundary components or the same component. This domain-dependent sign contradicts the uniform crossing relations of the HOMFLYPT skein algebra. We show that this discrepancy is resolved by passing to an anti-holomorphic orientation convention.

We establish all required orientation comparisons in the general setting of an oriented Lagrangian submanifold of a 6-dimensional symplectic manifold.

A local-to-global fixed point theorem for \CAT(0)! spaces with a uniform elliptic gap

• Authors: Jeroen Schillewaert
• arXiv: 2610.41468
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Metric Geometry (math.MG)

A finitely generated group acting locally elliptically on a complete \CAT(0) space of finite geometric dimension fixes a point if distances to elliptic fixed sets are uniformly controlled by displacements. For arbitrary groups, the same hypothesis gives a fixed point in the space or its visual boundary. The proof combines a first-moment random-walk displacement estimate with Izeki's zero-drift theorem. For finite-rank $\mathbb{R}$-buildings whose Euclidean metric factor has finite linear image, we obtain an all-type local-to-global theorem, including the nondiscrete case, with an explicit gap and without metric completeness; the linear-image restriction can be removed from the fixed-point conclusion when the Euclidean factor has dimension at most three. A tangent-cone criterion characterizes the gap and yields a cubical bound with optimal asymptotic growth $d/\pi$ in dimension $d$. For Hadamard manifolds, the gap is equivalent to uniformly bounded finite stabilizers in the isometry image, with stabilizer bounds of optimal polynomial growth in the gap constant.