math.GT daily digest: 11 new submissions for 26 August 2026

New submissions

Spectral Sequences in Lasagna Theory

• Authors: Amey Joshi
• arXiv: 2608.23909
• Subjects: Geometric Topology (math.GT)

We use Khovanov-Floer theories and their spectral sequences to construct differentials on the corresponding skein-lasagna modules. Then we show that for 2-handlebodies, a spectral sequence of TQFTs coming from a Khovanov-Floer theory gives a spectral sequence of lasagna modules under the aforementioned differential. As a corollary, we recover the rank inequality between the Khovanov and Lee lasagna modules. We also prove certain properties like the connect-sum formula that these spectral sequences satisfy.

Every Plat Presentation Admits a Positive Bounded Braid Representative

• Authors: Seth Hovland
• arXiv: 2608.24003
• Subjects: Geometric Topology (math.GT)

Classifying link types via their n-bridge positions requires navigating vast Hilden double coset classes H2n\B2n/H2n. We provide a restriction on this by proving that every Hilden double coset admits a positive braid representative whose Dehornoy minimum is explicitly bounded. The proof relies on the key observation that , which allows us to utilize the Garside left-greedy normal form to remove all powers of $Delta$ from any given braid representative. We provide an explicit upper bound on its Dehornoy minimum in terms of the Garside length of the initial presentation. This bounded representative restricts the algebraic space of plat presentations, providing a potential computational tool for studying bridge isotopy classes and offering a constructive step toward algebraic approaches to the Bridge Finiteness Conjecture.

Splitting singular fibers with periodic monodromies and their monodromy factorization

• Authors: Hyunggi Kim
• arXiv: 2608.24208
• Subjects: Geometric Topology (math.GT)

A Lefschetz fibration is a smooth 4-manifold admitting a surface bundle structure over a surface except at finitely many singular fibers, whose singularities are only of nodal type. From the structure of the singular fibers, the monodromy of each singular fiber is given by a right-handed Dehn twist along a curve, called a vanishing cycle, in the fiber. Collecting all monodromy data from the singular fibers, we obtain a monodromy factorization into right-handed Dehn twists, which completely determines the Lefschetz fibration.

In this paper, we construct a fibration with one singular fiber whose monodromy is periodic (that is, the monodromy homeomorphism is isotopic to a periodic map). Following the idea of Matsumoto, we give a splitting of the singular fiber into Lefschetz fibers and determine their vanishing cycles for a collection of periodic monodromies. We describe the construction of the splitting singular fibers and the procedure for reading vanishing cycles using two branched covering structures of the fibers. We also give splittings of singular fibers into Lefschetz fibers related to a family of periodic actions and determine their vanishing cycles.

Spectral geometry of Khovanov Laplacians

• Authors: Jernej Grlj, Aaron D. Lauda
• arXiv: 2608.24298
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic--combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.

Poisson bracket of trace functions of Atiyah-Bott-Goldman symplectic structure on the Fuchsian locus of the character variety

• Authors: Deblina Das, Arpan Kabiraj
• arXiv: 2608.24317
• Subjects: Geometric Topology (math.GT)

We develop a systematic method for computing traces of products of Möbius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we compute Poisson brackets of trace functions arising from geodesic representatives. As applications, we recover Wolpert's cosine and sine formulas.

Legendrian simple knots not detected by Khovanov homology

• Authors: Chun-Sheng Hsueh, Marc Kegel, David Suchodoll, Annika Thiele
• arXiv: 2608.24353
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

In this short note, we exhibit a Legendrian simple knot that is not detected by Khovanov homology. This disproves a conjecture of Chernov and Maguire.

Kwasik--Schultz manifolds are -compactifiable

• Authors: Shijie Gu
• arXiv: 2608.24416
• Subjects: Geometric Topology (math.GT)

Kwasik and Schultz constructed two-ended open $4$-manifolds which satisfy the usual finiteness and stability conditions at infinity but do not admit arbitrarily small $1$-neighborhoods. In particular, neither end is collarable, so the manifolds are not completable. We show that the open manifolds associated to their non-desuspendable $C\_2$-actions nevertheless admit finite-dimensional compact ANR -compactifications. Consequently, there exists a -compactifiable open $4$-manifold which is not pseudo-collarable. This answers a question of Guilbault and Tinsley.

Realising automorphisms of the extended intersection form by diffeomorphisms

• Authors: Csaba Nagy
• arXiv: 2608.24617
• Subjects: Geometric Topology (math.GT)

Suppose that is a normal -smoothing of a $2q$-manifold $M$ over some , where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or ``Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism . Assuming that is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if is also free, then we show that every automorphism can be realised. These results are obtained by studying a ``two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author.

As an application, we show that for a complex $q$-dimensional complete intersection, with even, every automorphism of the cohomology ring is realised by a diffeomorphism.

The Smooth Narrow Mordell-Weil Group of Elliptically Fibered 4-Manifolds

• Authors: Maria Morariu
• arXiv: 2608.24673
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

For an elliptically fibered 4-manifold , we introduce a subgroup of the smooth mapping class group of $M$ with explicit geometric representatives: the smooth narrow Mordell-Weil group . Leveraging Kodaira's classification of singular fibers, we construct a natural isomorphism between and the orthogonal complement of the trivial lattice in . This identification parallels its holomorphic counterpart. This paper generalizes work of Farb and Looijenga, who defined the smooth Mordell-Weil group and computed it for fibrations over a sphere with only nodal fibers. We give an explicit formula for the rank of depending only on the genus of the base curve and the types of singular fibers of $pi$. As a corollary, we show that rational elliptic surfaces are the only elliptic surfaces with holomorphic and smooth Mordell-Weil groups of the same rank.

Cross submissions

A sharp hyperbolic volume bound for hypersurfaces in

• Authors: Lizhi Chen, Kuntao Jin
• arXiv: 2608.24057
• Cross-list: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

Let be a closed oriented hyperbolic three-manifold normalized so that . We prove a sharp lower bound for the volume of hypersurfaces in representing the slice class \[M^3 \\times \\{ \\mathrm{pt} \\}\] \\in H\_3(M^3 \\times \\mathbb{S}^1; \\mathbb{Z})!, and we classify the equality case. If $g$ is a smooth Riemannian metric on with the scalar curvature , then every closed embedded hypersurface $Sigma$ representing the slice class \[M^3\\times\\{\\mathrm{pt}\\}\]! satisfies . The bound is attained by the product metric , with $h$ any metric on . Conversely, if equality holds for some $Sigma$, then up to a diffeomorphism preserving the slice class, and .

Computing an e-net of a closed hyperbolic surface

• Authors: Vincent Delecroix, Vincent Despré, Camille Lanuel, Hugo Parlier, Monique Teillaud
• arXiv: 2608.24497
• Cross-list: cs.CG
• Subjects: Computational Geometry (cs.CG); Geometric Topology (math.GT)

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips.

The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part.

As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.