math.GT daily digest: 12 new submissions for 27 August 2026

New submissions

The three-dimensional symplectization question for overtwisted contact structures

• Authors: Nobuo Iida
• arXiv: 2608.24942
• Subjects: Geometric Topology (math.GT)

We consider closed connected oriented three-manifolds equipped with positive cooriented contact structures. We prove that a symplectomorphism between their symplectizations forces the existence of an orientation-preserving diffeomorphism between the underlying three-manifolds under which the two contact structures are homotopic as oriented plane fields. Combining this formal rigidity with the classification of overtwisted contact structures, we show that two closed overtwisted contact three-manifolds have symplectomorphic symplectizations if and only if they are contactomorphic. Thus, we resolve the three-dimensional symplectization question when both contact structures are overtwisted.

Obstructing Unknotting Number Two For Some Alternating Knots

• Authors: Justin Gebel, William Prangley
• arXiv: 2608.25256
• Subjects: Geometric Topology (math.GT)

We apply the $d$-invariants of Ozsváth-Szabó and a theorem due to Brendan Owens to determine the unknotting numbers of 194 alternating knots with 13 or fewer crossings.

Link Floer homology, nonformality, and the Borromean rings

• Authors: Kristen Hendricks, Matthew Stoffregen, Ian Zemke
• arXiv: 2608.25295
• Subjects: Geometric Topology (math.GT)

Our main result is a computation of the full link Floer complex of the Borromean rings. The Borromean rings are well known to be an L-space link. We prove that the full link Floer complex is not a free-resolution of its homology, in contrast to the situation for L-space knots, plumbed L-space links, and two-component L-space links. We also describe the link involution of the Borromean rings, up to a minor ambiguity. The proof goes by way of studying some general techniques about -deformations of L-space link Floer complexes. Our techniques give a simple criterion for when an L-space link must have formal link Floer complex, which reproves the existing formality results for one- and two-component L-space links.

A condition of admissibility for generalized Alexander quandles

• Authors: Katsunori Arai, Keisuke Himeno, Ryoya Kai, Yuko Ozawa
• arXiv: 2608.25345
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)

A typical example of a quandle is the conjugation quandle. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. We study the admissibility problem for quandles, that is, determining whether a given quandle is admissible. In particular, we focus on generalized Alexander quandles, which are groups equipped with quandle structures defined by group automorphisms. In this paper, we provide a sufficient condition for admissibility in terms of geometric properties of quandles: namely, if every antipodal set in each algebraically connected component consists of a single point, then the quandle is admissible. This result generalizes previous results on generalized Alexander quandles.

A diagrammatic grading bound for colored link homology

• Authors: Hongjian Yang
• arXiv: 2608.25676
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We establish a grading bound for colored link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston--Bennequin number of a Legendrian link from link homology, in the spirit of Ng.

The half-signature extension of the 3D bordism bicategory and its representations

• Authors: Glen Lim
• arXiv: 2608.25680
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

We construct the half-signature bordism bicategory as a central extension of the 3-dimensional oriented bordism bicategory by $\mathbb{Z}$. This is an index $2$ subextension of the signature bordism bicategory. Following methods of Bartlett--Douglas--Schommer-Pries--Vicary, we construct a presentation of this bicategory and hence show that its linear representations are classified exactly by modular tensor categories.

Gröbner Bases for Alexander Fitting Ideals of Links

• Authors: Takefumi Nosaka, Kotaro Shimizu
• arXiv: 2608.25725
• Subjects: Geometric Topology (math.GT)

Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander--Gröbner invariants of links. Over $\mathbb{Q}$, for the maximal free-abelian coefficient system of a connected compact oriented $3$-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine \AG_2! for a two-component pretzel family.

Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds

• Authors: Shintaro Fushida-Hardy, Robert Harris, B. Doug Park
• arXiv: 2608.25889
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the Bogomolov-Miyaoka-Yau line in the complex geography plane. We also discuss how the conjectural symplectic Bogomolov-Miyaoka-Yau inequality implies that our results are close to being optimal.

The kernel of the Birman-Craggs-Johnson homomorphism

• Authors: Tara E. Brendle, Dan Margalit, Andrew Putman
• arXiv: 2608.26001
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For surfaces of genus with at most one boundary component, we give explicit generating sets for the kernel of the Birman-Craggs-Johnson homomorphism and the commutator subgroup of the Torelli group. As an application, we give a new proof of Johnson's calculation of the abelianization of the Torelli group.

All once-extended 3D TQFTs are Reshetikhin--Turaev theories

• Authors: Glen Lim
• arXiv: 2608.26068
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

We present a direct geometric construction which extends the Reshetikhin--Turaev TQFT to circles. We prove that this agrees with the generators-and-relations approach of Bartlett--Douglas--Schommer-Pries--Vicary, thus providing an alternative to the Cerf-theoretic part of their classification of once-extended 3-dimensional TQFTs, and deduce as a result that all once-extended 3-dimensional TQFTs arise via our construction.

Cross submissions

Spin structures and measures on the moduli space of curves

• Authors: Paul Norbury
• arXiv: 2608.25237
• Cross-list: math.AG
• Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Geometric Topology (math.GT)

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space \cal M}_{g,n}^{\rm spin! of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

Linear isoperimetric filling inequalities in Hadamard spaces at and above the asymptotic rank

• Authors: Jonas W. Peteranderl
• arXiv: 2608.26059
• Cross-list: math.MG
• Subjects: Metric Geometry (math.MG); Group Theory (math.GR); Geometric Topology (math.GT)

Reformulated in terms of the asymptotic rank, a conjecture by Gromov predicts a linear isoperimetric filling inequality in all dimensions greater than or equal to the asymptotic rank of a Hadamard space, in contrast to the Euclidean-type nonlinear behavior below this threshold. We prove the predicted linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank. More precisely, every integral cycle of dimension at or above the asymptotic rank admits a filling whose mass is bounded linearly in the mass of the cycle. Our proof is based on a new self-improvement mechanism for Wenger's sub-Euclidean growth theorem. This approach upgrades the asymptotic rank-one result by Wenger and the recent asymptotic rank-two result by Lang, Stadler, and Urech from exponents arbitrarily close to one to the optimal linear exponent. Moreover, the result extends to arbitrary finite asymptotic ranks.