math.GT daily digest: 20 submissions for 15 September 2026

New submissions

Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant

• Authors: Zining Fan
• arXiv: 2609.13326
• Comments: 18 pages, 5 figures
• Subjects: Geometric Topology (math.GT)

We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let be a link with link components , and let be the conjugation quandle of the dihedral group . We use the semidirect-product structure to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus $n$. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix . The abelian group , which we call the signature module, is represented by an integer matrix . We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of , are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all . The all-reflection system is the same as the Fox-$n$ colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups $\mathrm{Dih}(B)$ for every abelian group $B$.

Stratification and realization of singularity data on the Loch Ness monster

• Authors: Nate Fisher; Camilo Ramírez Maluendas
• arXiv: 2609.13607
• Comments: 21 pages, 8 figures
• Subjects: Geometric Topology (math.GT)

The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface.

We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data.

These results provide the first realization theorem for strata of non-compact tame translation surfaces and reveal a strong interaction between singularity data and the ends space.

Brown's Asymptotic Limit Functor and Proper Homology

• Authors: Sumanta Das; Rekha Santhanam
• arXiv: 2609.14063
• Comments: 63 pages. Comments welcome!
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits and : it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown--Grossman proper fundamental group of any open contractible $3$-manifold other than is uncountable and perfect.

Torus knots as loop-edges in three-sphere triangulations

• Authors: Lezhi Lin; Jonathan Spreer
• arXiv: 2609.14200
• Comments: 12 pages, 6 figures, 2 tables
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus.

The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.

On Nilpotent and Hypocentral Quandle Homomorphisms

• Authors: Yuki Imamura; Tomoki Yoshida
• arXiv: 2609.14270
• Comments: 46 pages
• Subjects: Geometric Topology (math.GT); Category Theory (math.CT); Group Theory (math.GR)

We introduce a notion of nilpotency for surjective quandle homomorphisms, which relativizes the nilpotency of quandles recently defined by Darné. A surjective homomorphism is called nilpotent if its relative inner automorphism group is nilpotent relative to the inner automorphism group of the source. We show that this condition can equally be described through the relative transvection group and the relative adjoint group, and that it is equivalent to being a finite composite of covering homomorphisms.

Extending the relative lower central series to transfinite ordinals, we then introduce hypocentral homomorphisms and characterize them as the surjections that are right orthogonal to the strongly connected homomorphisms. As a consequence, the strongly connected and the hypocentral homomorphisms form an orthogonal factorization system for the surjections in the category of quandles.

Finally, we also introduce reduced and operator-reduced homomorphisms, which relativize reduced quandles, and give sufficient conditions for such a homomorphism to be nilpotent.

Weights of finite cyclic actions on definite $4$-manifolds

• Authors: David Baraglia
• arXiv: 2609.14328
• Comments: 44 pages
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with . Let $p$ be a prime and suppose that acts smoothly on $X$ (if we also require an assumption on how $G$ acts on ). Using equivariant Yang--Mills theory, Hambleton--Lee and Hambleton--Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of on which $G$ acts linearly. We give a new proof of this result using equivariant Seiberg--Witten theory. Furthermore we also determine the weights of all equivariant line bundles on $X$, showing that these likewise coincide with that of an equivariant connected sum of linear actions on .

Gluing $\partial$-Morin maps on manifolds with boundary and its applications

• Authors: Koki Iwakura
• arXiv: 2609.14403
• Comments: 36pages, 7 figures, 1 table
• Subjects: Geometric Topology (math.GT)

We study $\partial$-Morin maps, a class of smooth maps from manifolds with boundary such that both the maps themselves and their restrictions to the boundary have only Morin singular points, and the singular point sets of the maps are disjoint from the boundary. We develop a gluing construction that provides a unified framework for studying maps from manifolds with boundary through corresponding maps from closed manifolds. Using this framework, we derive Euler characteristic formulas for $\partial$-Morin maps and a congruence relating the signature to the self-intersection number of the singular point set for $\partial$-fold maps from $4$-manifolds to $3$-manifolds. We also establish an inequality relating singular fibers of stable maps from a $3$-manifold to the plane to the simplicial volume of the source manifold. We further apply these results to the existence problem for $\partial$-fold maps and to the non-singular extension problem.

Unknotting number and L-space satellite operations

• Authors: Daren Chen; Ian Zemke; Hugo Zhou
• arXiv: 2609.14520
• Comments: 58 pages
• Subjects: Geometric Topology (math.GT)

We give a lower bound for the unknotting number under the L-space satellite operations. In the case of braided L-space satellite operations, including cabling operations, we obtain a stronger bound. The proof uses a lower bound for the unknotting number in knot Floer homology due to Alishahi-Eftekhary. The computational tool is a formula for computing the knot Floer complex under the L-space satellite operations, which was defined in the authors' previous work.

Delta-Unknotting Number for Montesinos Knots

• Authors: Kazumichi Nakamura
• arXiv: 2609.14522
• Comments: 29 pages, 16 figures, 2 tables
• Subjects: Geometric Topology (math.GT)

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. In this paper, we determine the $\Delta$-unknotting numbers for certain families of Montesinos knots. Using results on pretzel knots and two-bridge knots, we prove that for these families the $\Delta$-unknotting number equals the absolute value of the second coefficient of the Conway polynomial. In particular, every positive pretzel knot belongs to these families, and certain two-bridge knots also belong to them.

From veering triangulations to convergence actions and back again

• Authors: Jason Fox Manning; Saul Schleimer; Henry Segerman
• arXiv: 2609.14598
• Comments: 90 pages, 43 figures
• Subjects: Geometric Topology (math.GT)

Suppose that $M$ is a finite-volume cusped hyperbolic three-manifold, equipped with a veering triangulation $\mathcal{V}$. We prove that the action of the fundamental group of $M$ on the veering two-sphere is a geometrically finite convergence action. Applying a result of Yaman, we deduce that the veering two-sphere is equivariantly homeomorphic to the boundary of hyperbolic space.

As an application, we obtain Cannon-Thurston maps associated to veering triangulations. If $\mathcal{V}$ is layered we recover the classical Cannon-Thurston map. If it is not we obtain Cannon-Thurston maps that do not come from surface subgroups. These are the first such examples in the cusped case.

Finally, we implement an algorithm to draw approximations of these Cannon-Thurston maps. This improves upon previous approximations obtained by Thurston and others.

The Lorenz braid index and hyperbolic volume

• Authors: Thiago de Paiva; Connie On Yu Hui; José Andrés Rodríguez Migueles
• arXiv: 2609.14931
• Comments: 28 pages, 4 figures. This is an expanded version of some sections of arXiv:2410.04391v2, the mathematical content of which has been re-organised into two preprints with better focuses and exposition: arXiv:2410.04391v3 and this article. An additional result is included
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

The homology cobordism group is not generated by graph homology $3$-spheres

• Authors: Ben Mares; Yuta Nozaki; Masaki Taniguchi
• arXiv: 2609.15415
• Comments: 12 pages, 1 figure, 1 table. Comments are welcome
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We show the homology cobordism group of homology $3$-spheres is not generated by graph homology $3$-spheres. Our technique is a combination of filtered instanton Floer theory and Gromov's simplicial volume.

On possible values of the signature of flat unitary bundles over compact surfaces

• Authors: Inkang Kim; Pierre Pansu; Xueyuan Wan
• arXiv: 2609.15549
• Comments: 61 pages, comments are welcome
• Subjects: Geometric Topology (math.GT)

We determine all possible signatures of flat Hermitian bundles over compact, connected, oriented surfaces of positive genus with nonempty boundary. If has genus and boundary components, then the signatures arising from representations are exactly the integers $m$ satisfying [ |m| \leq (p+q)(2g+n-2) -(2g-2)|p-q| -\min{2,n|p-q|}. ] Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging $p$ and $q$, in .

Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three

• Authors: Longsong Jia
• arXiv: 2609.15602
• Comments: 26 pages; comments welcome
• Subjects: Geometric Topology (math.GT); Complex Variables (math.CV)

Let be a connected complex totally geodesic submanifold. We prove that if , then $N$ is the image of an entire Teichmüller space under a covering construction induced by a finite branched cover of marked surfaces. This answers a question of Arana-Herrera and Wright.

The Dax isomorphism for topological $4$-manifolds

• Authors: Jianfeng Lin; Yi Xie; Boyu Zhang
• arXiv: 2609.15792
• Comments: 16 pages; comments are welcome!
• Subjects: Geometric Topology (math.GT)

Let $X$ be an oriented topological $4$-manifold with boundary. We establish a Dax isomorphism for the fundamental group of the space of embedded arcs in $X$. Here, the embedding space can be either the simplicial set of locally flat embeddings or the space of topological embeddings endowed with the compact-open topology. When $X$ is smooth, we show that the space of smooth arcs and topological arcs have canonically isomorphic fundamental groups, and the topological Dax isomorphism agrees with the smooth Dax isomorphism. As a consequence, the homomorphisms that arise in the smooth Dax isomorphism do not depend on the smooth structure.

PL Recognition After Two Stabilisations is PSPACE-Hard

• Authors: Rhuaidi Antonio Burke
• arXiv: 2609.15890
• Comments: 16 pages, comments welcome
• Subjects: Geometric Topology (math.GT)

Let $Z$ be any fixed closed connected PL $4$-manifold. We give a polynomial-time many-to-one reduction from the compressed word problem in Thompson's group $F$ to fixed-target recognition of . Once a finite triangulation of $Z$ has been fixed, the construction sends a straight-line program $A$ to a closed triangulated PL $4$-manifold such that in $F$ if and only if is PL-homeomorphic to . Since the compressed word problem in $F$ is PSPACE-complete, every such recognition problem is PSPACE-hard. In particular, fixed-target recognition of is PSPACE-hard for every fixed .

Cross submissions

Topological classification through knotted graphs: Fermi surface dispersions and Lifshitz transitions

• Authors: Hakan Akgün; Xianquan Yan; Ching Hua Lee
• arXiv: 2609.13390
• Cross-list: cond-mat.mes-hall
• Comments: 50 pages total: 12-page main article (including references) and 38-page Supplemental Material; 6 main-text figures and 20 supplemental figures
• Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Geometric Topology (math.GT); Computational Physics (physics.comp-ph)

Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

Cyclotomic expansions of colored $SU(n)$ invariants of two-strand torus knots and Bailey transforms

• Authors: Chuwen Wang
• arXiv: 2609.13766
• Cross-list: math.QA
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT); Representation Theory (math.RT)

Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen--Liu--Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots and give a Bailey theoretic realization of the cyclotomic coefficients. For and , the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \sum_{m=0}^{N} \left(\prod_{j=0}^{m-1}{N-j}{N+n+j}\right) H_m^{(n,p)}(q), $$ where is independent of the color $N$.

The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin--Zheng formula for then gives a well-poised Bailey kernel. A terminating very-well-poised summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra.

As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen--Liu--Zhu $SU(n)$ volume conjecture for .

Patterns in the Markov numbers and their generalizations

• Authors: Cormac O'Sullivan
• arXiv: 2609.14149
• Cross-list: math.NT
• Comments: 49 pages, 18 figures
• Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

Positive integer solutions to correspond to the important Markov (Markoff) numbers when . From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real $D$. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual case is part of a bigger picture. Encoding function graphs are shown to be convex for , straight lines for and concave for . Among other consequences are generalizations to all $D$ of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.

On Correspondences between the Alexander Polynomials of Special Alternating Links and MOY Graphs

• Authors: Leonardo Rodrigues de Medeiros; Arker Oke Soe
• arXiv: 2609.14809
• Cross-list: math.CO
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

Fox's conjecture famously asserts that the absolute values of the coefficients of the Alexander polynomial of alternating links are trapezoidal. In the setting of MOY graphs, where a different notion of Alexander polynomial appears, the equivalent result to Fox's conjecture is known to hold. In this paper, we relate the two polynomials in the case of special alternating links. More precisely, we show that their degrees, first, and last coefficient agree, and that the MOY polynomial coefficients always dominate the classical Alexander polynomial.