math.GT daily digest: 12 submissions for 11 September 2026

New submissions

Bar cohomology of links: beyond Milnor invariants

• Authors: Greg Friedman; Nir Gadish; Robin Koytcheff; Dev Sinha; Ben Walter
• arXiv: 2609.11009
• Comments: 44 pages, 5 figures
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

A counterexample to the wrapping number conjecture

• Authors: Qiuyu Ren
• arXiv: 2609.11084
• Comments: 4 pages, 2 figures; the main result was obtained by GPT-6 Astra
• Subjects: Geometric Topology (math.GT)

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

Homological lifts of Arnold invariants and

• Authors: Noboru Ito
• arXiv: 2609.11427
• Comments: 29 pages, 3 figures
• Subjects: Geometric Topology (math.GT)

Viro's Euler-integral polynomial and the Lanzat--Polyak quantized-curvature polynomial refine Arnold's invariants and for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is . The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in . Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

Trace Coordinates and Local Fenchel--Nielsen Parameters in Real Hyperbolic 5-Space

• Authors: Krishnendu Gongopadhyay; Sagar B. Kalane; Abhishek Mukherjee
• arXiv: 2609.11438
• Subjects: Geometric Topology (math.GT)

We study representations of fundamental groups of closed orientable surfaces into $\mathrm{SL}(2,\mathbb{H})$, whose projectivization $\mathrm{PSL}(2,\mathbb{H})$ is the group of orientation-preserving isometries of real hyperbolic $5$-space, with the aim of developing a quaternionic analogue of Fenchel--Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is $15$-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there.

For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally $6$-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus $g$, this yields real parameters.

Commensurability Relations Between Deligne-Mostow-Thurston and Ghazouani-Pirio Monodromy Groups

• Authors: Chenglong Yu; Yiming Zhong
• arXiv: 2609.11482
• Comments: 11 pages, 1 figure
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

We classify commensurability relations between Deligne--Mostow--Thurston monodromy groups and the sixteen arithmetic Ghazouani--Pirio monodromy groups arising from moduli of flat cone metrics on the sphere and on the torus. In both settings, the monodromy group preserves a skew-Hermitian form coming from twisted homology. We use a degeneration method to compute their determinant classes in the genus one case. The defining CM fields and determinant classes of the skew-Hermitian forms determine the commensurability relations. In particular, each of the sixteen arithmetic Ghazouani--Pirio monodromy groups is commensurable with an arithmetic Deligne--Mostow--Thurston monodromy group.

Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of

• Authors: Anar Akhmedov
• arXiv: 2609.11727
• Comments: 23 pages, 3 figures
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)

We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in , we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees.

We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of . Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups.

We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes and in the nontrivial -bundles over and , respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology . Suitable gluings in the twisted $(2,4)$ case have , , and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of . We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology , obtained via knot surgery and twisted fiber sums.

Symplectic Surface Summing and Negative Spheres

• Authors: Anar Akhmedov
• arXiv: 2609.11803
• Comments: 10 pages, 1 figure
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We extend the sphere-summing construction of \cite{AZ} from torus sums of spheres to symplectic sums along surfaces of arbitrary genus. The relative surfaces may have arbitrary positive intersection number with the summing surface. We give formulas for the genus and self-intersection of the resulting surface, together with graph and simultaneous versions of the construction. The original elliptic-surface case is recovered as a special case, and higher-genus examples are obtained from hyperelliptic Lefschetz fibrations and braided symplectic surfaces in .

The Chen-Yang volume conjecture for long integral fillings of fundamental shadow links

• Authors: Ce Shen
• arXiv: 2609.11839
• Comments: 51 pages
• Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA)

We prove the Chen--Yang volume conjecture for all sufficiently long integral Dehn fillings of any fixed marked fundamental shadow-link exterior. For each fixed filling, the exponential growth of its $SO(3)$ Turaev--Viro invariants recovers its hyperbolic volume along the full sequence of odd levels. The filling coefficients may have mixed signs and unrelated magnitudes. We also establish a complete asymptotic expansion of the signed $SO(3)$ Witten--Reshetikhin--Turaev invariant and identify the absolute leading coefficient explicitly in terms of adjoint Reidemeister torsion. Fixed even colors on the filling cores recover characters of the geometric holonomy. The key difficulty is cancellation in the signed surgery sum. Our main analytic tool transfers an exact reflection symmetry from a continuous model to the finite quantum sums. We control the error below the exponential scale of the surviving contribution. We also apply the method to a one-edge state sum restricted to central colors. The dominant contributions cancel, and for each sufficiently large fixed number of blocks we determine the smaller surviving exponential rate and its nonzero leading coefficient.

Cross submissions

D-modules and Solvable Lie Foliations

• Authors: Ameth Ndiaye
• arXiv: 2609.10583
• Cross-list: math.RT
• Subjects: Representation Theory (math.RT); Differential Geometry (math.DG); Geometric Topology (math.GT)

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the -module and the derived ring . We compute the D-irregularity $\mathrm{D}\text{-}\mathrm{irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of , establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.

On Diagrammatic Categorification of Verma Modules I: Braiding

• Authors: Pedro Guicardi
• arXiv: 2609.10941
• Cross-list: math.QA
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT); Representation Theory (math.RT)

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of . Our motivation is to construct a theory of Khovanov homology for knot complements in (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of , thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in , which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

• Authors: Masaharu Ishikawa; Tat-Thang Nguyen
• arXiv: 2609.11396
• Cross-list: math.AT
• Comments: 16 pages, 1 figure
• Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); Combinatorics (math.CO); Geometric Topology (math.GT)

For every and , we construct a homogeneous polynomial of degree whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree , implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

A closed signature formula for the Katz-Long-Moody Hermitian form

• Authors: Haru Negami
• arXiv: 2609.11793
• Cross-list: math-ph
• Subjects: Mathematical Physics (math-ph); Geometric Topology (math.GT); Representation Theory (math.RT)

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible.

The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.