math.GT daily digest: 15 submissions for 2 October 2026

New submissions

Braided Surfaces and Small Exotic 4-Manifolds

• Authors: Anar Akhmedov
• arXiv: 2610.00409
• Comments: 26 pages, 3 figures
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

Let $K$ be the trefoil or figure-eight knot and let denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover [

M_K\times S^1\longrightarrow T^2\times S^2 ] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided -section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over . We then track the induced peripheral data and gluing diffeomorphisms in the constructions of and its twisted double , as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to \mathbb{CP}^2\#k\CPb^2!, \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.

skein algebras of small surfaces

• Authors: Cole Ellison, Vijay Higgins, Mark Sandey, Eden Sylvester
• arXiv: 2610.01075
• Comments: 30 pages
• Subjects: Geometric Topology (math.GT)

We study the skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in We compute the kernels of natural maps between the and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the skein algebra of the torus, which depends on the specialization of the quantum parameter $q.$

Colored Khovanov homology theories at prime roots of unity

• Authors: You Qi, Louis-Hadrien Robert, Joshua Sussan, Emmanuel Wagner
• arXiv: 2610.01122
• Comments: 23 pages, many figures and diagrams. Comments welcome!
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We equip Khovanov's colored $\mathfrak{gl}(2)$-homology groups with a degree-2 derivation. Over a field of characteristic , the derivation specializes to a $p$-nilpotent differential, which leads to a categorification of colored $\mathfrak{gl}(2)$-homology at a $2p$th root of unity.

A monodromy relation for the Cartwright-Steger fibration

• Authors: Anar Akhmedov, Sai-Kee Yeung
• arXiv: 2610.01467
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT)

We study the genus-$19$ Albanese fibration of the Cartwright--Steger surface and its order-three symmetry. Passing to the orbifold quotient of the elliptic base gives a relation among the two handle monodromies and the three Dehn twists about the vanishing cycles. The two real vanishing paths give cycles whose real fixed points lie on different ovals of the invariant fiber. For a cyclic triple cover, we express the intersection of a curve with its image under the deck transformation as a signed crossing count on the quotient. We determine the two local branch values of a degree-$72$ bicanonical pencil near a node and study a symmetric bicanonical pencil. A general pencil also gives a finite degree-$72$ map from a blowup of the surface to the product of the elliptic curve and a projective line. The monodromy translates of the nodal vanishing classes span the first homology of a smooth fiber.

Some petal diagrams of the unknot are hard

• Authors: Alexei Vernitski
• arXiv: 2610.01596
• Subjects: Geometric Topology (math.GT)

A petal diagram of a knot is a projection with a single multi-crossing and no nested loops; it is encoded by a permutation of the heights of the strands through the multi-crossing. Colton, Glover, Hughes and Sandberg proved a Reidemeister-type theorem for petal diagrams: two petal permutations represent the same knot if and only if they are related by trivial petal additions and deletions and by crossing exchanges. We ask whether every petal diagram of the unknot can be reduced to the one-petal diagram without ever increasing the number of petals, in analogy with Dynnikov's monotonic simplification theorem for rectangular diagrams. By an exhaustive, certified computer search we show that this is true for diagrams with at most 7 petals and false for 9 petals. Of the 40320 petal diagrams with 9 petals, 24992 represent the unknot, and exactly 108 of them are hard: none of them admits a crossing exchange or a trivial petal deletion, even if two natural petal-number-preserving symmetries are allowed. Up to these symmetries and mirror image there are three hard diagrams. Two of them can be untangled by passing through 11 petals; the third cannot be untangled through diagrams with at most 11 petals, but can through 13. We explain why the phenomenon differs from the rectangular case: a petal diagram is an arc presentation whose cyclic order of pages is determined by the order of its vertices on the binding, and no elementary move of Cromwell and Dynnikov preserves this rigid structure.

Ivanov's finite-index Frattini subgroup conjecture for subgroups of the Torelli group

• Authors: Renaud Detcherry
• arXiv: 2610.01734
• Subjects: Geometric Topology (math.GT)

Let be a closed oriented surface of genus $g$. Ivanov conjectured that if $G$ a finitely generated subgroup of its finite-index Frattini subgroup is nilpotent. We prove this conjecture for finitely generated subgroups of the Torelli subgroup , using recent partial results of the author about the Andersen-Masbaum-Ueno conjecture. We also show that the full AMU conjecture would imply Ivanov's conjecture.

Sections, pseudosections, and multisections of Lefschetz fibrations

• Authors: R. Inanc Baykur, Noriyuki Hamada
• arXiv: 2610.01776
• Comments: 38 pages, 18 figures in color
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We show that not every symplectic Lefschetz fibration over the $2$--sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus , which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

Complex hyperbolic surfaces whose closed geodesics are all simple

• Authors: Sami Douba, D. B. McReynolds, Julien Paupert, Jonathan Vittore
• arXiv: 2610.01806
• Comments: v1. 8 pages
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Number Theory (math.NT)

We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.

webs at roots of unity

• Authors: Vijay Higgins, Bryan Silva, Amy Somers, David Stephens, Srijan Sundar, Parth Wokhlu
• arXiv: 2610.01881
• Comments: 26 pages
• Subjects: Geometric Topology (math.GT)

We study webs on surfaces for the skein theory specialized at roots of unity. We construct the analogue of Chebyshev polynomials and use them to construct central elements in the skein algebra. We then focus on the cases of the 3-punctured sphere and the closed torus. We compute the Frobenius homomorphism for these two surfaces and show that the Frobenius homomorphism for the torus is not injective when the quantum parameter is a root of unity of even order $n.$

Cross submissions

Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials

• Authors: Honghuai Fang, Tian Zhou
• arXiv: 2610.00303
• Cross-list: math.QA
• Comments: 32 pages, 1 figure
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)

We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in . These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.

Mock Modularity, Resurgence, and Dual False Theta Functions

• Authors: Mrunmay Jagadale
• arXiv: 2610.00516
• Cross-list: hep-th
• Comments: 18 pages
• Subjects: High Energy Physics - Theory (hep-th); Geometric Topology (math.GT); Number Theory (math.NT)

The invariant of a three-manifold , a half-index of the $3d$ theory , is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce $q$-series, and making sense of the operation remains a central open problem. Seifert manifolds, whose invariants are built from false theta functions, offer a natural testing ground. We study ``dual false theta functions'', the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel--Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer $p$ satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge . We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product .

Ascending, Bridge, and Unknotting Numbers of Knotted Graphs

• Authors: Bahar Acu, Puttipong Pongtanapaisan, Ritikaa Singhal
• arXiv: 2610.00788
• Cross-list: math.CO
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

In this paper we define the ascending number and bridge number for virtual and welded spatial graphs. We prove that the former admits a lower bound in terms of the latter. This relationship allows us to compare different notions of unknotting virtual spatial graphs by crossing changes.

Pullback metrics and homological obstructions to BLD remetrization of branched covers

• Authors: Deguang Zhong
• arXiv: 2610.01087
• Cross-list: math.MG
• Comments: This is a revised version of the paper "A DEGREE OBSTRUCTION TO BLD REMETRIZATION OF BRANCHED COVERS" (originally posted on ResearchGate)
• Subjects: Metric Geometry (math.MG); Geometric Topology (math.GT)

We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of , , have uniformly porous branch values of Hausdorff dimension but admit no compatible Ahlfors $n$-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.

Cluster structures from Legendrian double twist knots: a comparative approach

• Authors: James Hughes, Jiajie Ma
• arXiv: 2610.02086
• Cross-list: math.SG
• Comments: 46 pages, 29 figures; comments welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces

• Authors: Ming Hong Tee
• arXiv: 2610.02171
• Cross-list: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

The Ahlfors--Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.