math.GT daily digest: 22 submissions for 9 September 2026

New submissions

Aspherical manifolds with nonvanishing tautological classes

• Authors: Mauricio Bustamante
• arXiv: 2609.06345
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

For every even integer m≥2, we construct a closed, orientable, smooth, aspherical (2m+1)-manifold whose fundamental group has nontrivial center and for which infinitely many tautological classes in the F2-cohomology of the classifying space of its group of homotopically trivial diffeomorphisms are nonzero and not nilpotent. Specializing the calculation to cohomological degree 0 gives examples of such manifolds that do not bound compact smooth manifolds. These provide counterexamples to a conjecture of Hebestreit--Land--Lück--Randal-Williams.

Locally Minimal Bridge Presentations of Knots

• Authors: Chad Musick
• arXiv: 2609.06492
• Comments: 20 pages, 11 figures, 4 tables. Comments are very welcome
• Subjects: Geometric Topology (math.GT)

For classical knots, many characteristics are difficult to determine from arbitrary diagrams of the knot. Bridge number is of this type. It is possible to have a bridge presentation of a knot in which the number of bridges is only locally minimal, as shown by Ozawa and Takao. Here, we give a method for beginning with an arbitrary classical knot diagram having \(n\) crossings and producing a locally minimal bridge presentation of the knot with size \(O(n^2)\). This method requires only polynomial time and space. For some classes of knots, any local minimum is also the global minimum, which was shown for the trivial knot by Otal. Because the unknot is the only knot with bridge number \(1\), unknot recognition is in \(P\).

Unknotting orientable surfaces

• Authors: Anthony Conway
• arXiv: 2609.06544
• Comments: 19 pages
• Subjects: Geometric Topology (math.GT)

It is shown that every locally flatly embedded genus g∈{1,2} surface in the 4-sphere with knot group Z is unknotted. The same proof establishes that any two genus g∈{1,2} surfaces in D4 with knot group Z and common boundary an Alexander polynomial one knot are isotopic rel. boundary. In previous work, the author and Powell reduced such unknotting problems to a question concerning the cancellation of (−t)-quadratic forms over Z[t±1], which was solved in genus g≥3 using work of Bass. In genus g=2, we observe that the same proof goes through using further work of Bass. In genus g=1, the result instead follows from a statement in commutative algebra which was proved with the assistance of AI. Combined with earlier work of Freedman on locally flat spheres with knot group Z, this shows that a locally flatly embedded orientable surface in S4 is unknotted if and only if its knot group is Z.

Bridge numbers of virtual graphs and handlebody-links

• Authors: Nikolaos Chantis; Puttipong Pongtanapaisan
• arXiv: 2609.06860
• Subjects: Geometric Topology (math.GT)

We study bridge indices of graphs, virtual graphs, and their handlebody classes. We construct virtual ravels for which the difference between bridge index and weighted overpass bridge index is unbounded. The lower bound uses exponentially many colorings for one fixed flow in a $\mathbb{Z}\_2$-family of biquandles. For each negative Euler characteristic we construct classical almost unknotted graphs of one fixed trivalent graph with arbitrarily large bridge index. We also obtain unbounded even bridge indices for ravels of each fixed bouquet rank. A separate almost unknotted family has an unbounded difference between its spatial-graph bridge index and the bridge index of its regular neighborhood. Finally, edge connected sums of Suzuki curves give exact constrained and unconstrained bridge indices whose difference is unbounded.

Bounded cohomology and optimal separating constant for representations

• Authors: Yushi Nakano; Teruhiko Soma
• arXiv: 2609.06937
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT)

Let Σ be a closed oriented surface of genus >1 and M a complete hyperbolic 3-manifold with a marking i:Σ⟶M. We consider the case that M has no parabolic cusps and at least one of the two ends is simply degenerate. For Γ=π1(Σ), let ρM:Γ⟶PSL2(C) be the holonomy of M and ρ:Γ⟶PSL2(C) any representation in PSL2(C). We will show that, if ρ is discrete and non-faithful, then

∥[Vol(ρ)]−[Vol(ρM)]∥∞≥v3

holds, where [Vol(ρ)] denotes the bounded fundamental class of ρ in the bounded cohomology H3b(Γ,R) of Γ and v3 is the volume of a regular ideal 3-simplex in H3. As an application, we present a rigidity theorem for ρM in the set of representations ρ of Γ in PSL2(C) in terms of [Vol(ρ)]. The rigidity theorem implies that v3 is the optimal separating constant.

Knots, black holes, databases, and birthdays: Collision entropy of knot invariants

• Authors: Pedro Olivares-Sánchez; Edison Jessie Vázquez Gordillo; Radmila Sazdanović; Carlos Alfonso Ruiz Guido; Aldo Guzmán-Sáenz; Renato Osvaldo Salmerón-García; Ramiro López-Vázquez; Ernesto Lupercio
• arXiv: 2609.08298
• Comments: 12 pages, 2 figures. This arXiv version contains the main article only
• Subjects: Geometric Topology (math.GT); Probability (math.PR)

A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If Dn and D′n are independent n-crossing diagrams, then Pr{ΔK(Dn)=ΔK(D′n)}=O(n−1/2), and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli(1/4) variables whose sum is a binomial coordinate in the 3-adic determinant valuation. Consequently, supa≥1Pr{detK(Dn)=a}=O(n−1/2). For a discrete invariant I, let αn(I)=Pr{I(Dn)=I(D′n)}. Its collision entropy is H2(I(Dn))=−logαn(I). An independent sample of size M has (M2)αn(I) colliding pairs on average and birthday scale αn(I)−1/2. If M2nαn(I)→∞, a repeated value occurs with probability tending to one even when αn(I)→0. For the determinant, M=o(n1/4) suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy 1−O(n−1/2) but cannot distinguish inequivalent pairs in one fiber.

Cusp cross-sections of low-dimensional arithmetic hyperbolic manifolds

• Authors: Marcus Grimbert; Duncan McCoy; Connor Sell
• arXiv: 2609.08756
• Subjects: Geometric Topology (math.GT)

We study which compact flat manifolds of dimensions 3≤n≤6 occur as cusp cross-sections in commensurability classes of non-compact arithmetic hyperbolic (n+1)-manifolds. Using the classification of low-dimensional flat manifolds together with the relationship between their holonomy representations and rational quadratic forms, we determine the possible arithmetic commensurability classes for every flat manifold in these dimensions. We show that every flat n-manifold for n=3,4,5 occurs as a cusp cross-section in infinitely many distinct arithmetic commensurability classes. In dimension six, the same holds with exactly eight exceptions: eight non-orientable flat 6-manifolds occur in a unique arithmetic commensurability class. We also show that, for every 3≤n≤6, there is a single commensurability class of arithmetic hyperbolic (n+1)-manifolds containing every flat n-manifold as a cusp cross-section.

Asymptotically conformal and asymptotically rigid mapping class groups

• Authors: Javier Aramayona; George Domat; Christopher Leininger
• arXiv: 2609.08849
• Comments: 40 pages, 5 figures
• Subjects: Geometric Topology (math.GT); Complex Variables (math.CV); Group Theory (math.GR)

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group H(S), has finite index in the asymptotically conformal modular group Mod0(X), where X is a hyperbolic structure on S. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in H(S) of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if S has n≥3 ends, then Mod0(X) has type Fn−1 but not FPn. We also establish analogous results for Lp modular groups.

Global rigidity of sphere packings in constant curvature background geometry

• Authors: Zunwu He; Guangming Hu; Ziping Lei
• arXiv: 2609.08856
• Subjects: Geometric Topology (math.GT)

In this paper, we study the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of hyperbolic, Euclidean, spherical and ideal hyperbolic sphere packings on a tetrahedron. Moreover, we establish the extended variational principle to obtain the global rigidity of the (ideal) hyperbolic sphere packings on 3-manifolds. We also generalize the conformal tetrahedra to higher dimensions and provide a characterization of conformal n(≥3)-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.

Three formulas for the Ising and Potts invariants of a knot or link

• Authors: Tristan Bullock; Thomas Kindred
• arXiv: 2609.09093
• Comments: 12 pages, 6 figure, 2 tables, comments welcome!
• Subjects: Geometric Topology (math.GT)

Jones described how the Ising and Potts models from statistical mechanics give rise, with appropriate choices of Boltzmann weights, to invariants of an oriented link. The Boltzmann weights that Jones proposed, however, work only with a correction factor that he does not mention. We fill in the missing details in two different ways. We also show that all of these invariants, which we call the Ising and Potts invariants, are given by evaluating the Jones polynomial at t equal to one of these Boltzmann weights.

Cross submissions

Tame polynomial automorphisms

• Authors: Stéphane Lamy
• arXiv: 2609.05655
• Cross-list: math.AG
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

The group of polynomial automorphisms of the affine n-space is an interesting large group. A slightly simpler group is its subgroup of tame automorphisms. Natural problems about these groups include the existence of normal subgroups, the classification of finite subgroups, the Tits alternative, and the possible dynamical degrees of their elements. One method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we introduce in detail. This paper is a survey focusing on the following three cases: dimension 2, dimension 3, and dimension 4 for tame automomorphism preserving a nondegenerate quadratic form.

Commensurability and quasi-isometry classification for one vertex one loop tubular groups

• Authors: Amy Tao
• arXiv: 2609.05705
• Cross-list: math.GR
• Comments: 29 pages
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

A tubular group has a graph of groups decomposition with Z2 vertex groups and Z edge groups. This paper gives a classification of one vertex one loop tubular groups G(k,ℓ),(m,n)=⟨a,b,t:[a,b]=1,tambnt−1=akbℓ⟩ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

Mutation Sequences along Weaves and Amalgamation of Braid Varieties

• Authors: Yuma Mizuno
• arXiv: 2609.05833
• Cross-list: math.AG
• Comments: 45 pages
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT); Representation Theory (math.RT)

Let p,q be positive braids with Demazure products u,v. The endpoint stratum Conf(p,q)u,v of the double Bott-Samelson cell splits as Conf(u,v)×X(p)×X(qop), a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on Conf(p,q), and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map X(p)×X(ΔΔ)→X(pΔ), where Δ is a reduced positive braid for w0 and dem(p)=w0.

Property Pnaive for non-orientable big Mapping Class Groups

• Authors: Jesús Hernández Hernández; Tianyi Lou
• arXiv: 2609.06110
• Cross-list: math.GR
• Comments: 16 pages, 5 figures
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We give a criterion for property Pnaive for subgroups of orientable big mapping class groups and apply it to mapping class groups of non-orientable infinite type surfaces through the orientation double cover.

Hierarchical geometry and right-angled Artin groups in graph braid groups

• Authors: Byung Hee An; Sangrok Oh; Jihoon Park
• arXiv: 2609.06589
• Cross-list: math.GR
• Comments: 48 pages, 16 figures. Comments are welcome!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

For the unordered discrete configuration space UDn(Γ) of n particles on a connected finite graph Γ, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, orthogonality, and product regions in the resulting hierarchically hyperbolic group (HHG) structure admit explicit descriptions in terms of configuration-space geometry, and we show that this structure satisfies the additional properties needed for constructing and obstructing subgroups isomorphic to right-angled Artin groups (RAAGs).

Using sufficiently subdivided models, we apply this hierarchy to graph braid groups. We give a finite combinatorial formula for the maximal rank of a free abelian subgroup and show that every RAAG occurs as an undistorted subgroup of some graph braid group. For graph 2-braid groups, we obtain stronger restrictions: every RAAG subgroup has bipartite defining graph, and the embedding problem is characterized by an induced-subgraph condition in the expanded core graph of the hierarchy. For the RAAG defined by the four-vertex path, this condition is equivalent to a finite graphical criterion on the underlying graph.

Lusternik-Schnirelmann category of Lee Forms on locally conformally symplectic manifolds

• Authors: Kenji Fukushi
• arXiv: 2609.06709
• Cross-list: math.SG
• Comments: 16 pages
• Subjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT); Geometric Topology (math.GT)

The Lusternik--Schnirelmann category of a closed symplectic manifold admits various estimates. Locally conformally symplectic (LCS) geometry is a generalization of symplectic geometry involving a closed one-form, called the Lee form. In this paper, we introduce Farber's Lusternik--Schnirelmann category for closed one-forms into LCS geometry by applying it to the Lee class. In particular, this provides a new topological approach to LCS structures of the second kind. We then obtain lower bounds for this invariant under LCS blow-ups.

Minimal Two-Spheres and Manifolds with Positive Isotropic Curvature

• Authors: Tsz-Kiu Aaron Chow; Yipeng Wang
• arXiv: 2609.06910
• Cross-list: math.DG
• Comments: 7 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We improve the Micallef--Moore index estimate for harmonic two-spheres in n-manifolds with positive isotropic curvature to the sharp bound n−3. Combining with recent work, this completes the diffeomorphism classification of closed manifolds with positive isotropic curvature in the remaining dimensions five and six.

Legendrian Reidemeister moves for the convex surface projection

• Authors: Zijian Rong
• arXiv: 2609.07123
• Cross-list: math.SG
• Comments: 10 pages + references. Comments welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We prove a Legendrian Reidemeister theorem for Legendrian knots in thickened convex surfaces.

Möbius Invariant Dimers and Miquel Dynamics

• Authors: Niklas C. Affolter; Luis Mühlhoff
• arXiv: 2609.07485
• Cross-list: math-ph
• Comments: 16 pages, 9 figures
• Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS); Geometric Topology (math.GT)

As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.

Crossed-module crossed braided categories

• Authors: Azat M. Gainutdinov; Ingo Runkel; Bangxin Wang
• arXiv: 2609.07626
• Cross-list: math.CT
• Comments: 60 pages
• Subjects: Category Theory (math.CT); Geometric Topology (math.GT); Quantum Algebra (math.QA)

For a crossed module χ:G→H, we introduce the notion of χ-crossed braided (resp. ribbon) categories, where the categories are graded by group G and carry an H-action. Our definition unifies and generalises several familiar notions: taking χ=id:G→G with the conjugation action recovers G-crossed braided categories; taking χ:G→{∗} for abelian G yields G-graded braided categories; taking χ:{∗}→G leads to braided categories equipped with a G-action. The equivalence relation between χ-crossed braided categories is typically finer than that between G-crossed braided ones. We classify χ-crossed braided structures on the category of G-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with G- and H-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to χ-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally χ-crossed ribbon or admit an orthogonal G-decomposition.

Geometric finiteness in paracomplex hyperbolic spaces

• Authors: Tianqi Wang; Zhufeng Yao; Tengren Zhang
• arXiv: 2609.07628
• Cross-list: math.DG
• Comments: 60 pages, comments are welcomed
• Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

We develop a framework for studying discrete subgroups of PGL(d+1,R) via the paracomplex hyperbolic space Hdτ, a rank-1 pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of Hdτ. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

Classification of Legendrian doubles and suspensions

• Authors: Yasemin Yildirim
• arXiv: 2609.08557
• Cross-list: math.SG
• Comments: 25 pages, 4 figures. Comments are welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.