math.GT daily digest: 21 submissions for 25 September 2026

New submissions

Smoothing topological isotopy of surfaces in 4-manifolds with a boundary geometric dual

• Authors: Jae Choon Cha; Byeorhi Kim
• arXiv: 2609.28598
• Comments: 12 pages
• Subjects: Geometric Topology (math.GT)

We prove that topological isotopy implies smooth isotopy for neat surfaces with common nonempty boundary in a smooth orientable 4-manifold, in the presence of a geometric dual in the boundary. Together with our earlier work on surface smoothing, this shows that the natural map from smooth isotopy classes to topological isotopy classes of such surfaces with prescribed boundary is bijective.

On symplectic aspects of $SU(2)$ character varieties for punctured surfaces

• Authors: Aliakbar Daemi; Christopher Scaduto
• arXiv: 2609.28907
• Comments: 76 pages, 12 figures
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold. When the moduli space is nonempty, there is a natural homomorphism from the mapping class group of the punctured surface to the symplectic mapping class group of this moduli space. It is shown that this homomorphism is injective if and only if the dimension of the moduli space is greater than $2$. This generalizes work of Seidel and Wehrheim--Woodward. Also given is a complete classification of Lagrangian spheres in the projective plane blown up at $5$ points with its monotone symplectic structure, which is the moduli space for the 5-punctured sphere. Furthermore, it is determined when two such Lagrangian spheres can be displaced by a symplectic isotopy. Results are also obtained regarding Lagrangian spheres in the intersection of two quadrics in . The proofs involve instanton Floer theory and results on Heegaard splittings. A main technical result establishes the approximation of any Hamiltonian isotopy of the $SU(2)$ moduli space by holonomy perturbations which are used in instanton homology.

Asymmetric L-space knots with arbitrary braid index

• Authors: Keisuke Himeno; Masakazu Teragaito
• arXiv: 2609.28917
• Comments: 10 pages, 8 figures
• Subjects: Geometric Topology (math.GT)

The first examples of asymmetric L-space knots were found by Baker and Luecke. Among Baker-Luecke knots, the simplest one has braid index 12. Later, it turned out that there are just 9 asymmetric hyperbolic L-space knots in the SnapPy census, and their braid indices take the values 4, 5, 6 and 7. It is known that asymmetric L-space knots have braid index at least 4. Recently, Baker and the second author give infinitely many asymmetric hyperbolic L-space knots with braid index 4.

In this paper, we construct an asymmetric hyperbolic L-space knot with arbitrary braid index bigger than 4. In fact, there exist infinitely many such knots for each braid index.

Projective Braids in

• Authors: Aakash Gupta; Louis H. Kauffman; Rama Mishra
• arXiv: 2609.29058
• Comments: 18 pages, 21 Figures
• Subjects: Geometric Topology (math.GT)

Links in projective space can be represented by diagrams in where the projective plane is represented by a disk with antipodal identifications on the boundary. In this context if we place an $n-$strand braid $\beta$ on this disk keeping its end points on the boundary then due to the identification of antipodal points it naturally represents a link diagram in . We call this the projective closure of the braid $\beta$. In this paper we show that not all links in possess a diagram represented by projective closure of some braid.

Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings

• Authors: Airi Aso
• arXiv: 2609.29124
• Comments: 14 pages, 4 figures
• Subjects: Geometric Topology (math.GT)

Let $K$ be a hyperbolic knot and let be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with detect the complex volume of the knot complement.

We compute $$ \frac{\pi}{2} \log \left( \frac{A_{K,n-2}(1)A_{K,n+2}(1)} {A_{K,n}(1)^2} \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatorname{Vol}(S^3\setminus K) +i,2\pi^2\operatorname{CS}(S^3\setminus K) $$ modulo . Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.

Quandle coloring quivers of pretzel links

• Authors: Qinghui Meng; Ximin Liu; Boxin Zhou
• arXiv: 2609.29127
• Comments: 33 pages,14 figures
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle . First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.

Adjoint Reidemeister torsion from hyperbolic gluing equations

• Authors: Shuang Ming; Baojun Wu
• arXiv: 2609.29299
• Comments: 32 pages,3 figures
• Subjects: Geometric Topology (math.GT)

Motivated by the study of asymptotics of quantum invariants, Dimofte and Garoufalidis introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic $3$-manifold. They proved that its constant term can be written in terms of Neumann-Zagier data and the complex shape parameters of ideal tetrahedra, and conjectured that it equals the adjoint twisted Reidemeister torsion. On the other hand, in the study of asymptotics of Turaev-Viro type invariants of cusped $ 3$-manifolds, the authors, together with Liu, Sun and Yang, found that the one-loop terms of their asymptotic expansions could be written in terms of Gram matrices and decorated edge lengths of ideal tetrahedra. In this paper, we prove the conjecture of Dimofte and Garoufalidis and relate the one-loop term appearing in the Turaev-Viro type invariant to the adjoint twisted Reidemeister torsion.

Exotic knot traces and the epsilon-invariant

• Authors: Shreya Sinha; Allison Tsypin
• arXiv: 2609.29432
• Comments: 23 pages, 14 figures. Comments welcome!
• Subjects: Geometric Topology (math.GT)

Building on recent work of Hom and Wan, we construct additional examples of exotic knot traces, providing partial progress toward their conjecture. Our approach uses the immersed-curve techniques in Heegaard Floer homology developed by Chen and Hanselman.

Bending parameterization of geometrically finite hyperbolic manifolds

• Authors: Bruno Dular
• arXiv: 2609.29450
• Comments: 24 pages, 1 figure, comments are welcome!
• Subjects: Geometric Topology (math.GT)

We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism.

This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.

The three-dimensional symplectization question for small Seifert L-spaces and further classes

• Authors: Nobuo Iida
• arXiv: 2609.29534
• Comments: 21 pages
• Subjects: Geometric Topology (math.GT)

We solve the three-dimensional symplectization question for several examples. These include small Seifert fibered $L$-spaces, a family of Whitehead-link surgery $L$-spaces, , certain surgeries on the figure-eight knot, explicit families of Brieskorn spheres, and arbitrary finite connected sums mixing these manifolds. In particular, the result covers the Weeks manifold and infinitely many hyperbolic $L$-spaces. The main symplectic step is to construct Liouville cobordisms in both directions from a symplectomorphism of symplectizations. The resulting preservation of strong fillability and of the vanishing of the monopole Floer contact invariant, together with the comparison with the Heegaard Floer contact invariant, connected-sum formulas, and classification results for tight contact structures, proves the stated rigidity results.

Fibred realizations of the prism quandle in the standard four-sphere

• Authors: Michal Jablonowski
• arXiv: 2609.29839
• Subjects: Geometric Topology (math.GT)

We prove that both smooth homotopy four-spheres arising from the two framed surgeries on a section of a period-six mapping torus built from a quaternionic prism manifold with cyclic factor of order seven are standard. Their belt spheres give two smooth fibred two-knot realizations of the prism quandle of order fifty-six in the standard four-sphere. The knots have the same knot group and diffeomorphic exteriors but are inequivalent. Combined with the fibred-rigidity reduction, this yields exactly two fibred realizations in homotopy four-spheres, both standard. The proof combines a Whitehead-link model for the orbit orbifold, circle-action and surgery methods, cyclic branched-cover calculations, and a spin-bordism evaluation of a secondary invariant. This resolves the order-seven case in the classical prism-family realization problem.

One internal stabilization is enough for resolving self-referential tubes

• Authors: Qizheng You
• arXiv: 2609.29854
• Comments: 14 pages, 5 figures. Comments are welcome!
• Subjects: Geometric Topology (math.GT)

This paper shows that one internal stabilization is enough to resolve all the self-referential tubes on an embedding. As an application, this paper gives a complete isotopy classification for embedded surfaces with a common geometric dual in orientable -bundle over a surface, provided that the embedded surfaces have greater genus than the base surface of the bundle.

Pairings of combinatorial 1-cocycles with loops in knot spaces

• Authors: Butian Zhang
• arXiv: 2609.29946
• Comments: 40 pages, 22 figures. Comments are welcome!
• Subjects: Geometric Topology (math.GT)

We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\mathbb{Z}$-valued combinatorial 1-cocycles and a $\mathbb{Z}/2\mathbb{Z}$-valued 1-cocycle on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that and can pair nontrivially with bracket and half-bracket loops. We conjecture that over $\mathbb{Q}$, and their mod 2 reductions together with over $\mathbb{Z}/2\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in . Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in and . Using these criteria, we show that descends to a nontrivial 1-cohomology class in , while and descend to linearly independent nontrivial 1-cohomology classes in .

Multivariate Quandles as Groupoid Invariants

• Authors: Xerxes D. Arsiwalla; Louis H. Kauffman
• arXiv: 2609.30262
• Comments: 36 pages
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Category Theory (math.CT); Rings and Algebras (math.RA)

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid $\mathcal{G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings $\mathbb{Z}[\mathbb{Z}]$, the multivariate version involves modules over groupoid rings $\mathbb{Z}[\mathcal{G}]$. A manifestly categorical description is given in terms of $\mathbb{Z}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between $\mathbb{Z}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

Cross submissions

Noncongruence cusp forms, Veech groups and enumeration

• Authors: John Rached
• arXiv: 2609.28514
• Cross-list: math.NT
• Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.

Robert J. Daverman (1941-2026)

• Authors: Gerard Venema; Craig Guilbault
• arXiv: 2609.28515
• Cross-list: math.HO
• Comments: with appendices by John Ewing and Dusan Repovs
• Subjects: History and Overview (math.HO); Geometric Topology (math.GT)

The is an expanded version of a memorial article that is to appear in the Notices of the American Mathematical Society.

Toric Richardson Varieties and Slice Links

• Authors: Thomas C. Martinez; Matthew J. Tyler
• arXiv: 2609.28536
• Cross-list: math.CO
• Comments: 27 pages, 5 figures. Comments welcome!
• Subjects: Combinatorics (math.CO); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

Proof of the positive trace gap conjecture

• Authors: Nikolay Bogachev
• arXiv: 2609.29033
• Cross-list: math.GR
• Comments: 20 pages. Comments are very welcome!
• Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT); Number Theory (math.NT)

We prove that a lattice $\Gamma$ in or has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense.

Our method also gives a similar result for lattices in , for every : a lattice has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb{Q}$ that splits over $\mathbb{R}$. We also discuss spectral consequences and other applications of the results and techniques.

On Auroux's Luttinger-surgery uniqueness question

• Authors: Nobuo Iida
• arXiv: 2609.29561
• Cross-list: math.SG
• Comments: 8 pages
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We construct simply connected integral symplectic four-manifolds forming an exotic pair, with equal values of , , , and , but with symplectic Kodaira dimensions $-\infty$ and $2$. Since symplectic Kodaira dimension is invariant under Luttinger surgery, the two manifolds are not Luttinger-surgery equivalent. This gives a negative answer to Auroux's uniqueness question.

S-meandric Permutations and Tangency Polynomials

• Authors: Yury Belousov; Viktoriia Georgievskaia
• arXiv: 2609.30068
• Cross-list: math.CO
• Comments: 28 pages, 10 figures
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.

Derivative links in contact topology

• Authors: Joseph Breen; Alexander Zupan
• arXiv: 2609.30182
• Cross-list: math.SG
• Comments: 25 pages + references, 18 figures. Comments welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We import the theory of $R$-links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of Lagrangian and symplectic sliceness, and more generally establishes one approach to what we call the Lagrangian slice-ribbon conjecture. Our main theorem asserts that a Legendrian knot with Thurston-Bennequin invariant $-1$ is Lagrangrian slice (resp. regularly Lagrangian slice) if and only if it supports a tight transverse (resp. Legendrian) $R$-link derivative.