math.GT daily digest: 14 new submissions for 4 August 2026

New submissions

A bijective topological recursion for maps

• Authors: Gaëtan Borot; Alessandro Giacetto; Lasse Merkens
• arXiv: 2608.00117
• Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Combinatorics (math.CO)

We prove bijectively a recursive excision formula for maps of arbitrary topology. This yields the topological recursion formulae governing their enumeration, already at the level of formal generating series and without any analyticity assumption. We extend the result to maps carrying self-avoiding loop models and to stuffed maps, i.e. a variant of maps allowing faces of arbitrary topology. Our results give a precise combinatorial meaning to all terms of the (blobbed) topological recursion known for (stuffed) maps, including the recursion kernel, which had so far been missing. The construction relies on the simple idea of iterating Tutte's algorithm until the topology changes. Equivalently, it can be interpreted as a pair-of-pants decomposition driven by a path issuing from the root of the first boundary, providing an analogue for maps of the Mirzakhani-McShane identity for hyperbolic surfaces.

Naturality in real Heegaard Floer theory

• Authors: Gary Guth; Ciprian Manolescu
• arXiv: 2608.00256
• Subjects: Geometric Topology (math.GT)
• Comments: 127 pages, many figures. Comments welcome!

In a previous paper we defined real Heegaard Floer homology, an invariant of three-manifolds equipped with involutions. Here we prove that real Heegaard Floer homology is natural, and that it admits an action of the equivariant mapping class group. We also establish naturality of real link Floer homology and real sutured Floer homology, and give a definition of involutive real Heegaard Floer homology.

Multisections and bridge positions in arbitrary dimensions

• Authors: Sylvain Courte; Delphine Moussard; Qiuyu Ren; Xiaozhou Zhou
• arXiv: 2608.00460
• Subjects: Geometric Topology (math.GT)
• Comments: 27 pages, 9 figures in color

Multisections were defined by Ben Aribi--Courte--Golla--Moussard as a way to decompose closed manifolds into $1$-handlebodies, which is a generalization of Heegaard splittings and trisections. Previously, their existence was only known in dimensions up to $5$. We show that multisections exist for closed manifolds in arbitrary dimensions. We also show that submanifolds of codimension at least $2$ in multisected manifolds can always be put into an appropriate bridge position, generalizing the existence result on bridge trisections in dimension $4$.

An infinite family of strongly invertible L-space knots without Khovanov thin surgery

• Authors: Marc Kegel; Masakazu Teragaito
• arXiv: 2608.00864
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages, 9 figures

Currently, there exist only three strongly invertible L-space knots that are known to admit no Khovanov thin surgery. In this article, we give the first infinite family of such knots.

Quadrangulation moves

• Authors: Philipp Korablev
• arXiv: 2608.01044
• Subjects: Geometric Topology (math.GT)

We prove that any two quadrangulations of the same surface, i.e., decompositions of a surface into squares, can be transformed into one another by a finite sequence of two-dimensional local cubical Pachner moves and non-local subdivision transformations.

Discriminant Varieties for Stick Knots and Links

• Authors: Alexander Kolpakov; Igor Rivin
• arXiv: 2608.01277
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Algebraic Topology (math.AT)
• Comments: 16 pages; partially LEAN verified; GitHub repository this https URL

How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.

Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces

• Authors: Leyi Zhu
• arXiv: 2608.01444
• Subjects: Geometric Topology (math.GT)
• Comments: 28 pages, 7 figures, 2 tables

A seamless parametrization of a closed oriented surface carries a discrete invariant, its holonomy signature: the cone angles, all multiples of $pi/2$, together with the rotational holonomy of the induced cross field. This is the datum a quadrangulation prescribes, and it decides whether any parametrization exists at all. Shen, Zhu, Capouellez, Panozzo, Campen and Zorin asked which signatures occur and gave a sufficient condition of gcd type; which signatures are realizable has remained open. We answer the question. A Reduction Lemma shows that the mapping class group acts on signatures with fixed cone angles with orbits classified by the subgroup alone, so at most three cases survive per angle multiset instead of . A dictionary then identifies seamless parametrizations with meromorphic 4-differentials, under which measures primitivity, and realizability becomes non-emptiness of a stratum of primitive $k$-differentials with and . Unwinding this against the known classification of such strata leaves exactly five exceptional families; every other admissible signature is realizable, in every genus. Two of the five appear to be new, and both live in genus two. Four of the five lie outside the gcd condition, and the whole region it leaves open is settled here. The non-emptiness half is made constructive by an explicit one-vertex square-tiled surface in every genus together with a local surgery that splits one cone into two of prescribed angles, leaving the genus, the other cones and untouched. Two extensions follow: surfaces with boundary, the feature-aligned setting, and the relation to the Abel-Jacobi criterion at a fixed conformal structure.

Embedded surfaces with trivial extendable mapping class groups in simply connected $4$-manifolds

• Authors: Weizhe Niu
• arXiv: 2608.01504
• Subjects: Geometric Topology (math.GT)
• Comments: 84 pages, 3 figures. Comments welcome

For every , every closed, connected, oriented, simply connected smooth $4$-manifold $X$, and every knot , we construct infinitely many pairwise topologically inequivalent smoothly embedded oriented genus-$g$ surfaces whose orientation-preserving extendable mapping class subgroups are trivial in both the topological and smooth categories and whose first Alexander modules are isomorphic to the Alexander module of $K$. In particular, there are infinitely many such surfaces with vanishing first Alexander module. The construction is supported in a $4$-ball. Although their Alexander data are prescribed independently, the surfaces are distinguished, and their mapping-class rigidity is detected, by the nonabelian centralizer structure of their exterior groups.

2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings

• Authors: Montek Singh Gill
• arXiv: 2608.01532
• Subjects: Geometric Topology (math.GT)
• Comments: 16 pages

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form , where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but and . In the case of , the conjecture holds if we replace with . A spherical counterexample for the projective variant is the Poincaré homology sphere . In the case of , the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere .

Minimal Hyperbolic Area of Teichmuller Curves in Genus Two

• Authors: Xiaoyu Su; Yumin Zhong
• arXiv: 2608.01984
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
• Comments: 34 pages. Comments welcome

We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group \Delta(2,5,\infty). The proof combines a small-area classification of noncompact hyperbolic orbifolds with a derivative-preserving affine descent construction for nonsquare quadratic differentials. The three possible nonsquare zero patterns are then excluded by arithmetic, marked-point, and covering obstructions.

Cross submissions

Hex9: A Quasi-Authalic, Quasi-Continuous Hexagonal DGGS on the Reference Ellipsoid

• Cross-list: physics.geo-ph
• Authors: Ben Griffin
• arXiv: 2608.00022
• Subjects: Geophysics (physics.geo-ph); Geometric Topology (math.GT)
• Comments: 41 pages, 18 figures. Reference implementation and machine-verified enumeration/measurement scripts: this https URL (Python, tag: paper_v2) and this https URL (C/C++)

Discrete global grid systems are conventionally designed over a prior coordinate reference system and inherit its compromises. Hex9 inverts the direction of design: we ask what requirements a hierarchical grid must satisfy to be geometrically coherent -- intrinsic orientability, flat mode transport, vertex closure, refinement commutativity -- and show that these requirements essentially determine the grid. The admissible cell primitive is the triangle; the admissible seed is the octahedral triangulation of S^2; admissible refinement is uniquely aperture 9, and uniquely up to chirality, with exactly one orientation of the hexagonal dual lattice per chirality -- fixed by a machine-verified exhaustive enumeration. The surviving structure is a shifted-aperture-9 hexagonal hierarchy in which every cell carries a unique address derived from the construction alone: truncated at level L, the address is a DGGS zonal identifier in the sense of OGC Topic 21; carried to the limit, a function recovers from it a point on the reference ellipsoid to arbitrary precision. The addressing is quasi-continuous -- position is recoverable everywhere except on a measure-zero set of seams -- rather than continuous in the strict ISO 19111 sense; the same object serves as both zonal identifier and position-recovery coordinate, with no prior coordinate reference system as input. A separable geometric realisation -- an analytical octahedral base projection composed with an optimal-transport-derived area-correcting warp -- places the grid on WGS84 with quasi-uniform cell areas: at level 5, 99% of the 708,588 cells lie within 0.005% of ideal area, with residual deviation confined to the six octahedral vertices required by the topology. The combinatorial grid is projection- and ellipsoid-independent; only the warp is specific to the reference body, and is recomputable for any ellipsoid, terrestrial or planetary.

Periodic quasiflats in hierarchically hyperbolic spaces

• Cross-list: math.GR
• Authors: Pénélope Azuelos; Mark Hagen
• arXiv: 2608.01513
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Metric Geometry (math.MG)
• Comments: 65 pages, 1 figure

We prove a quasiflat closing theorem and a coarse flat torus theorem for hierarchically hyperbolic groups (HHGs). Namely, given an HHG $G$, we prove that $G$ is hyperbolic if and only if it contains no subgroups and, if is virtually , then there is an $A$--invariant $n$--dimensional uniform quality quasiflat $F$ such that any two points in $F$ are joined by a uniform-quality hierarchy path lying in $F$. The later is a consequence of a more detailed theorem describing a ``coarse minset'' for $A$ in $G$, which has various applications, including an ascending chain condition for virtually abelian subgroups, hierarchical quasiconvexity of highest abelian subgroups, and some geometric control over normalisers, centralisers, and commensurators of abelian subgroups. We use this to rule out HHG structures for certain Coxeter groups on the basis of their affine subgroups, and to give a new proof that virtually solvable subgroups of HHGs are virtually abelian, which simplifies the original proof by avoiding Gromov's polynomial growth theorem.

Monodromy action on character varieties for Lefschetz pencils

• Cross-list: math.AG
• Authors: Ishan Banerjee
• arXiv: 2608.01700
• Subjects: Algebraic Geometry (math.AG); Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 25 pages, 5 figures

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].

Word maps and surface relations in symmetric groups

• Cross-list: math.GR
• Authors: Ewan Cassidy
• arXiv: 2608.02210
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Representation Theory (math.RT)
• Comments: 32 pages, 6 figures, comments welcome!

We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $chi$ of the symmetric group $S\_n$ and with R\_{g}=\[a\_{1},b\_{1}\]\\dots\[a\_{g},b\_{g}\]! and , we compute \\mathbb{E}\_{S\_{n}^{2g}}\\left\[\\chi\\left(R\_{g}(h)\\right)\\#\\mathrm{fix}\\left(w(h)\\right)\\right\]!. We show that, if $w$ is a shortest representative for the conjugacy class of , then this expectation is . As an application, we recover a boundedness statement of Magee--Puder on the large $n$ limit of the expected number of fixed points of , where is fixed and is chosen uniformly at random.