math.GT daily digest: 13 new submissions for 11 August 2026

New submissions

Random Knots via Stiefel manifolds

• Authors: Alexander Kolpakov; Igor Rivin
• arXiv: 2608.07605
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Probability (math.PR)
• Comments: 12 pages; LEAN verified; GitHub repository

A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.

Cobordism groups of dihedral branched covers

• Authors: Valentina Bais; Alexandra Kjuchukova
• arXiv: 2608.07788
• Subjects: Geometric Topology (math.GT)
• Comments: 29 pages, 11 figures, 0 footnotes, 1 fruit

For every integer , we compute the cobordism groups of dihedral $n$-fold branched covers of with oriented and non-oriented branching sets. We show that the groups are cyclic, generated by the $n$-fold connected dihedral covers of $(2,n)$-torus links. The isomorphism types of the groups are detected using explicit cobordism invariants defined in terms of the Seifert forms on the branching sets, generalizing Cappell-Shaneson characteristic knots associated to dihedral covers.

-Analogue of Ismagilov's Theorem

• Authors: Stéphane Tchuiaga
• arXiv: 2608.07802
• Subjects: Geometric Topology (math.GT)

We establish a -analogue of Ismagilov's theorem on the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold, we prove that the first continuous cohomology of the identity component of the group of volume-preserving homeomorphisms, with coefficients in the Banach space of continuous zero-mean functions, is isomorphic to the first de Rham cohomology. The proof introduces a topological transport theory for volume-preserving isotopies, yielding a continuous volume flux homomorphism and an associated transport cocycle. As an application, we obtain a resolution of the volume-preserving -Flux Conjecture.

Generalizations of the groups : graphs, moduli spaces, algebraic geometry, spherical braids

• Authors: Vassily Olegovich Manturov
• arXiv: 2608.07976
• Subjects: Geometric Topology (math.GT)
• Comments: 9 pages

In this work, we construct a generalization of the -theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space and the hypergraph encoding projective constraints. In contrast to the original theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups.

Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.

Type Annihilation for Classifying Maps of Rack Spaces

• Authors: Takefumi Nosaka
• arXiv: 2608.08076
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)
• Comments: 20 pages, Key wards: rack, quandle, rack space, classifying map, associated group, rack homology, group homology, type of a rack

We study the classifying map c\colon BX\to K(\As(X),1)! of a rack $X$ of finite type. Let t = \Type(X)!. We prove that t c\_{n\*}=0! for every when $X$ is connected, and that t^{n-1}c\_{n\*}=0! on the torsion subgroup \Tor H\_n^\mathbb{R}(X)! without any connectedness assumption. For a finite rack, under our sign conventions, the rationalized classifying map in degree $n$ is given by times the canonical projection from the $n$-fold tensor power of the orbit module to its $n$-th exterior power. For an arbitrary rack of finite type, we determine H\_2^{\mathrm{gr}}(\As(X);\mathbb{Z}\[1/t\])!. We also derive low-dimensional applications to symplectic and Alexander structures.

Geometric Monodromy of Mixed Braid Groups and the Multivariate Burau Representation

• Authors: Athira E V; Pranav Haridas
• arXiv: 2608.08079
• Subjects: Geometric Topology (math.GT)
• Comments: 38 pages, 19 figures

We study the monodromy action of the mixed braid group on the first cohomology of cyclic branched covers of , which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.

A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras

• Authors: Jin-Cheng Guu
• arXiv: 2608.08441
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 12 pages

Let $\Sigma$ be a compact connected oriented surface of genus at least $3$ with at most one boundary component. Santharoubane associated to certain presentations of the mapping class group modulo its center a finitely presented algebra equipped with a canonical surjection onto the generic Kauffman bracket skein algebra of $\Sigma$, and conjectured that a suitable choice yields an algebra isomorphic to the skein algebra. We show that every algebra arising from this construction admits an augmentation character, whereas the generic skein algebra of $\Sigma$ admits no unital character over $\mathbb{Q}(A)$. The latter obstruction follows from the intersection-one Dehn-twist identity together with a $4$-holed-sphere skein relation. Consequently, the conjectured isomorphism does not hold as stated.

Growth Rate of Essential Surfaces via Measured Laminations

• Authors: Brevan Ellefsen
• arXiv: 2608.08833
• Subjects: Geometric Topology (math.GT)

We show that in any Haken 3-manifold $M$, the dimensions of $\mathcal{ML}(M)$ and $\mathcal{ML}\_0(M)$ can be calculated using normal surfaces. As a corollary, we prove the number of closed orientable essential surfaces of Euler characteristic at least $-2n$ in a closed orientable hyperbolic 3-manifold with is asymptotic to .

A refinement of the Q-Polynomial for twisted knots

• Authors: Tumpa Mahato; Prabhakar Madeti
• arXiv: 2608.09514
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Quantum Algebra (math.QA)
• Comments: 23 pages, 32 figures, 5 tables

This paper introduces a two-variable polynomial invariant for oriented twisted knots, denoted by , refining the $Q$-polynomial of N. Kamada and S. Kamada \cite{NaoSei}. We exhibit an infinite family of twisted knots indistinguishable by the $Q$-polynomial but separated by the -polynomial. To prove invariance, we first determine a generating set of oriented Reidemeister moves for twisted knot diagrams, extending the result of Ali \cite{Dan} for oriented virtual knots; this result is new and of independent interest, as it provides the minimal framework needed to verify invariance of any oriented twisted knot invariant. As further applications, we derive an explicit crossing change formula, obtain lower bounds on the Gordian distance between homotopic twisted knots, examine the existence of cosmetic crossings in a twisted knot diagram, and finally prove that is a Vassiliev invariant of order one.

Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions

• Authors: Jennifer Hom; Shunyu Wan
• arXiv: 2608.09734
• Subjects: Geometric Topology (math.GT)
• Comments: 14 pages, 1 figure

We show that the numerical invariants $\nu$ and $|\epsilon |$ coming from knot Floer homology are knot $n$-trace invariants for any integer $n$, resolving the remaining case in Hayden-Mark-Piccirillo. This extension allows us to construct new families of exotic pairs using Yasui patterns. Moreover, by studying the invariant , we give a new topological obstruction to a $4$-manifold being a strong symplectic filling of any contact structure on the boundary.

Cross submissions

Non-semisimple open-closed 3d TFT

• Cross-list: math.QA
• Authors: Ingo Runkel; Yilong Wang
• arXiv: 2608.08057
• Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Geometric Topology (math.GT)
• Comments: 52 pages

Given a spherical finite tensor category C, not necessarily semisimple, and a two-sided modified trace on its projective ideal, we define an open-closed three-dimensional topological field theory with values in vector spaces. The bordism category has as morphisms three-dimensional bordisms with corners, whose boundary is partitioned into the gluing boundary, parametrised by source and target surface, and the unparametrised free boundary. The free boundary is equipped with an embedded C-coloured graph satisfying an admissibility condition. Our construction starts from a new three-manifold invariant based on the multi-handlebody invariant of [arXiv:1809.07991] and the chromatic maps of [arXiv:2302.04509]. The open-closed topological field theory is then obtained via the universal construction.

Searching for $J$-holomorphic curves via machine: first steps

• Cross-list: math.SG
• Authors: James Rowan; Yuan Yao
• arXiv: 2608.08473
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
• Comments: 26 pages, 15 figures, comments welcome!

We assemble numerical algorithms to search for $J$-holomorphic curves in symplectic manifolds. Each algorithm employs several different numerical techniques, each technique addressing a different aspect of the geometric problem. We separately consider both classical Fourier expansion and deep neural networks in our algorithms and compare their performance. Our algorithms take as input a smooth curve in a given homology class and search for a $J$-holomorphic curve in the same homology class. We first verify we can produce explicitly known holomorphic curves in complex manifolds, for example the Weierstrass $\wp$ function on the torus and curves in with the standard complex structure. Then we search for $J$-holomorphic curves in with non-integrable almost complex structures: essentially we start with a known holomorphic curve in an integrable almost complex structure $J\_0$, deform $J\_0$ to a nearby nonintegrable almost complex structure $J\_ϵ$, and use our methods to find the nearby $J\_ϵ$-holomorphic curve.

Stein structures not determined by the contact boundary

• Cross-list: math.SG
• Authors: Nobuo Iida
• arXiv: 2608.09361
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
• Comments: 36 pages

We construct two Stein structures on the same compact smooth four-manifold whose boundary contact forms are strictly identified and whose first Chern classes agree, but whose exact symplectic forms are not symplectomorphic. Moreover, no diffeomorphism makes the associated Weinstein structures homotopic.

The examples are obtained by removing a tubular neighborhood of a smooth bicanonical curve from a fake projective plane and comparing the induced Stein structure with its conjugate. Their canonical Spinc structures differ by a nonzero class of order two. After normalizing the boundary forms using a common Boothby-Wang connection, a hypothetical symplectomorphism preserves the oriented circle fiber and extends over the divisor cap, contradicting canonical-class rigidity. For the Weinstein statement, the required fiber-preservation is obtained from a monopole Floer grading asymmetry, using the Nelson-Weiler computation and Taubes's ECH-Seiberg-Witten correspondence. This gives an affirmative solution to a normalized version of the relative filling problem following Problem 4.96 in the K3 problem list.