math.GT daily digest: 17 submissions for 20 October 2026

The Tuesday, 20 October 2026 arXiv math.GT new-submissions listing contains 17 eligible papers: 15 new submissions and 2 cross submissions. Replacement submissions are excluded.

New submissions

Explicit formula for the Rasmussen invariant of odd pretzel knots

Authors: Yury Belousov, Vadim Stepaniuk
arXiv: 2610.24083
Comments: 10 pages, 7 figures
Subjects: Geometric Topology (math.GT)

We compute the Rasmussen s-invariant for all odd pretzel knots, giving an explicit closed formula. The formula extends to all slice-torus invariants that are invariant under mutation. In the absence of mutation invariance, we prove the formula under a mild condition on the pretzel parameters, which we conjecture to be unnecessary.

Official arXiv abstract page

Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds

Authors: David Gabai, Lily Qiao Li
arXiv: 2610.24190
Comments: 27 pages, 17 figures, comments welcome
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We show that for any closed hyperbolic 3-manifold $M$ , there are only finitely many Anosov or pseudo-Anosov flows on $M$ up to orbit equivalence.

Official arXiv abstract page

On the kissing number of hyperbolic surfaces with a small systole

Authors: S. Barré, M. Pichot
arXiv: 2610.24222
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

Parlier proved that there exists a constant such that for any hyperbolic surface $S$ of genus $g$ with systole $ℓ$ the following holds: [ Kiss(S)\leq U \frac {e^\ell}\ell g ] where $K i ss (S)$ is the maximum number of homotopically distinct systoles in $S$ (Kissing number).
We show that there exists a constant and a sequence of hyperbolic homogeneous surfaces of genus with a bounded systole such that [ Kiss(S_n)\geq Vg_n ] for all . This shows that Parlier's theorem is sharp for a fixed systole length.
Our proof uses Bruhat--Tits buildings of type and Hamiltonian surfaces.

Official arXiv abstract page

Analytic quantum invariants

Authors: Rinat Kashaev, Vladimir Mangazeev, Marcos Marino
arXiv: 2610.24247
Comments: 69 pages, 6 figures
Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

Using a deformed Borel-Laplace resummation, we propose a framework for repackaging quantum invariants as analytic invariants. We illustrate our approach through a detailed analysis of the coloured Jones polynomials of several simple knots, deriving explicit expressions for the resulting analytic invariants in terms of $(q, X)$ -series. These invariants are closely related to, but generally distinct from, those obtained via the conventional Borel resummation of the Melvin-Morton-Rozansky expansion. As a further example, we construct an analytic invariant from a sequence of two-variable polynomials conjectured to be the Links-Gould polynomials of the figure-eight knot, associated with the quantum superalgebras . Finally, we apply our technique to $q$ -series arising in the study of three-manifold invariants, revealing connections with mock modular forms.

Official arXiv abstract page

Length-convexity and domination for Hitchin representations

Authors: Subhojoy Gupta
arXiv: 2610.24606
Comments: 15 pages, 1 figure
Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let $S$ be a closed oriented surface of genus . For any , let be the Hitchin component and let be the subset of representations that take values in if and if . In this note we prove that for any Hitchin representation there is such that $ρ$ dominates $j$ in the Hilbert length spectrum; this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for $n = 3$ . The key step is establishing a convexity of the Hilbert length function in the Bonahon-Dreyer parametrization of ; this uses a result from the recent work of Barman-Gupta. We also prove these convexity and domination results for translation length functions and their spectra.

Official arXiv abstract page

Density of length ratios on Teichmüller space

Authors: Nhat Minh Doan, Hanh Vo
arXiv: 2610.24700
Comments: 19 pages, 2 figures
Subjects: Geometric Topology (math.GT)

We study length ratios of simple closed geodesics on finite-type hyperbolic surfaces of positive genus. We allow cusps and fix the geodesic boundary lengths. Let $d$ be the dimension of the corresponding Teichmüller space . We prove that for every positive vector , the set of surfaces carrying distinct essential interior simple closed geodesics with lengths proportional to this vector is dense in . In particular, one-holed tori with three distinct simple closed geodesics of equal length are dense. The proof uses high powers of Dehn twists, asymptotics of length differentials, and the Brouwer fixed-point theorem.

Official arXiv abstract page

Stein fillable contact structures with half Giroux torsion

Authors: Alberto Cavallo, Irena Matkovič
arXiv: 2610.24922
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We show that there exist Stein fillable contact 3-manifolds which contain a separating half Giroux torsion layer. Our examples are on Seifert fibred spaces over the sphere.

Official arXiv abstract page

A counterexample to the geometric triangulation conjecture of Costantino, Frigerio, Martelli and Petronio

Authors: Ke Feng, Aijin Lin
arXiv: 2610.25217
Comments: 53 pages, 5 figures, 6 tables. Comments are welcome!
Subjects: Geometric Topology (math.GT)

Costantino, Frigerio, Martelli and Petronio conjectured that an ideal triangulation in which every edge has valence at least six can be realized by hyperbolic partially truncated tetrahedra on the compact manifold it defines. We show that this is false. Starting from the figure-eight knot complement, we construct a triangulation with $2688$ tetrahedra and edge valences six and eighteen whose manifold is hyperbolic with totally geodesic boundary, but which admits no such realization. The obstruction is a critical point of the covolume energy of Luo and Yang at which three tetrahedra have collapsed: convexity then excludes every geometric competitor, an Euler characteristic count excludes edges of length zero, and the critical point itself is located by a finite certificate in exact integer arithmetic. The obstruction persists in every finite cover, while a complementary theorem shows that no counterexample can have only two edge lengths or only two orbits of edges.

Official arXiv abstract page

Negative-Definite Plumbings and New Unknotting Numbers

Authors: Seong-Jin Lee
arXiv: 2610.25430
Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Algebraic Topology (math.AT)

We use Neumann's normal form to decide whether the double branched cover of a knot bounds a negative-definite plumbing tree in either orientation, starting from the recognised Seifert or graph-manifold structure of the cover. For many non-alternating knots with at most thirteen crossings whose unknotting numbers were unknown, this identifies the cases in which the Heegaard Floer homology of the cover can be computed from such a plumbing, and shows for most of the others that it cannot. We determine 112 new unknotting numbers. For 43 of them, the lower bound rests on Heegaard Floer homology computed from the plumbing; for the other 69, it comes from linking pairings and from a theorem of McCoy and Park on special alternating knots. Finally, we show that the knot 13n1587 is a further explicit counterexample to the Bernhard--Jablan conjecture, after 13n3370.

Official arXiv abstract page

Biquandle Image Quivers

Authors: Geunyoung Kim, Sam Nelson, Puttipong Pongtanapaisan
arXiv: 2610.25593
Comments: 11 pages
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We introduce a new family of weighted quiver-valued invariants of oriented classical and virtual knots and link, surface-knots and surface-links, and immersed surface-knots and surface-links associated to finite biquandles. We derive several polynomial invariants from these quivers via decategorification. We provide examples to illustrate the computation of the invariant as well as toy examples showing tables of invariant values for small biquandles.

Official arXiv abstract page

Which Immersions into Euclidean Space are Boundary Immersions?

Authors: Austin Konkel
arXiv: 2610.25618
Comments: 27 pages, 16 figures. This paper here is a condensed portion of the author's PhD thesis in progress. A.I. tools were used only to help create figures and to survey for existing literature. Comments are encouraged!
Subjects: Geometric Topology (math.GT)

Here we study the problem of whether a given codimension one immersion $f$ of a closed and connected manifold into Euclidean space can be extended to an equidimensional immersion $F$ of some compact manifold $M$ whose restriction to the boundary parameterizes the image $S$ of $f$ . Such an extension $F$ exists if and only if (i) simple conditions on orientations and wrapping numbers are satisfied and (ii) a distinguished finite subset of the first cohomology of $S$ with coefficients in a certain sheaf of finite groups contains the preferred point.

Official arXiv abstract page

Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Authors: Atsuhide Mori
arXiv: 2610.25878
Comments: 15 pages. Companion to arXiv:2609.30406
Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Symplectic Geometry (math.SG); Quantum Physics (quant-ph)

This paper is a non-planar counterpart of the author's earlier work on planar contact structures and topological quantum computation. There the page is planar and all branch points are paired by disjoint liftable arcs, compatibly with the joint eigenlines of the Ising representation and with Calabi-Yau (CY) Stein fillings, those with vanishing first Chern class. Here the page has genus g and connected binding, presented as a simple degree-three cover of the disk. Every maximal system of disjoint liftable arcs has exactly g members and leaves two branch points unused. The half-twist interchanging the two points is not liftable, but its cube is, and lifts to the identity. An integral residue criterion supplies the CY test. A factorization localized on a contact connected sum is automatically CY. Unlike in the planar case, a CY factorization need not be the shortest. Once the fusion channels of the g pairs are fixed, Ising fusion determines the channel of the remaining pair, and braiding acts there only by scalars. Fibonacci fusion can leave that channel undetermined, and the powers of the cube then act by ten distinct relative phases on a single joint eigenspace, while the CY Stein filling stays fixed

Official arXiv abstract page

Topological mapping class groups of non-simply connected 4-manifolds

Authors: Jae Choon Cha
arXiv: 2610.26400
Comments: 37 pages, 4 figures
Subjects: Geometric Topology (math.GT)

We construct topological isotopy invariants of homeomorphisms of non-simply connected 4-manifolds using the topological Dax invariant of disks and the Freedman-Quinn invariant of spheres. Our invariants are effective for various non-smoothable 4-manifolds without assuming that the fundamental groups are good. They detect infinite-rank free abelian direct factors of the pseudoisotopy-trivial and homotopy-trivial mapping class groups, and give lower bounds on the Prüfer rank of the Torelli groups modulo homotopy-trivial mapping classes, including examples of infinite Prüfer rank. Our method is also useful for smooth 4-manifolds. We construct stably smoothable homeomorphisms of the Enriques surface and many Dolgachev surfaces that act trivially on homology but are not homotopic to the identity.

Official arXiv abstract page

The Structure of the Ghost and Swapping Algebras

Authors: Caleb Ashley, Ming Hong Tee
arXiv: 2610.26726
Subjects: Geometric Topology (math.GT)

The swapping algebras and the ghost algebras were introduced in the papers of Labourie and Bridgeman--Labourie to encode the Poisson structure on spaces of length functions and generalized cross-ratio functions associated to $Θ$ -Anosov representations. They used these algebras to show that correlation functions are stable under the Poisson bracket associated to Atiyah-Bott-Goldman's symplectic form and that length functions are convex along the Hamiltonian flow of other length functions. In this paper, we study the algebraic relationship between these two algebras. In particular, we show that the kernel of a natural surjection from the ghost algebra onto the extended swapping algebra is generated by five families of relations. Consequently, we obtain an isomorphism from an explicit quotient of the ghost algebra onto the extended swapping algebra.

Official arXiv abstract page

Uniform convergence of canonical discrete uniformization on equilateral approximations

Authors: Aijin Lin, Chunlei Liu
arXiv: 2610.26798
Comments: 50 pages
Subjects: Geometric Topology (math.GT)

We prove uniform convergence of the canonical discrete uniformization maps for every sequence of equilateral polygonal disks whose mesh tends to zero and whose marked Jordan boundaries converge uniformly. The limit is the normalized Riemann map, with convergence on the entire closed reference triangle. This proves Conjecture 7.3 of Luo, Sun and Wu for the original construction with Delaunay flips. The proof combines closed-domain compactness, marked lattice limits, a packing and resistance estimate for logarithmic cusp scales, and an envelope comparison principle, followed by a local affine energy argument.

Official arXiv abstract page

Cross submissions

Asymptotics for the number of saddle connections on a slit torus

Authors: David Aulicino, Howard Masur
arXiv: 2610.24429
Cross-list: math.DS
Comments: 16 pages, 2 figures
Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

We consider a flat torus with a marked line segment $β$ and prove linear asymptotics for closed curves on the torus disjoint from the interior of the line segment. We also consider segments that begin and terminate at the end points of $β$ whose interiors are disjoint from $β$ and prove $L lo g L$ asymptotics. In all cases, after quotienting by the growth rate, we compute the constant explicitly, which differs depending on whether or not $β$ lies in a periodic direction on the torus. In particular, when $β$ lies along a non-periodic trajectory, we show the constant is . Moreover, we prove that as a function of the holonomy vector of $β$ , the constants are continuous at the points where $β$ lies in a non-periodic direction.

Official arXiv abstract page

The minimum number of vertices of a triangulation of

Authors: Xinrui Chang (Fudan University)
arXiv: 2610.25171
Cross-list: math.CO
Comments: 9 pages; ancillary files include verification code and covering certificates
Subjects: Combinatorics (math.CO); Algebraic Topology (math.AT); Geometric Topology (math.GT)

We give a computer-assisted proof that every finite simplicial triangulation of real projective five-space has at least twenty-four vertices. Together with the construction of Guyer, Steinerberger, and Yang, this determines the minimum. A cup-product argument produces an induced subcomplex on sixteen or seventeen vertices. Transfer to the spherical double cover then imposes a covering condition on its lifted faces. In the sixteen-vertex case, a direct capacity bound excludes twenty-two vertices. For twenty-three vertices, a cochain extension argument reduces every relevant completion to one of 115 critical signed complexes; integer weights and exact covering certificates exclude all of them. The argument requires no assumption that the triangulation is combinatorial.

Official arXiv abstract page

The Tuesday, 20 October 2026 arXiv math.GT new-submissions listing contains 17 eligible papers: 15 new submissions and 2 cross submissions. Replacement submissions are excluded.

New submissions

Explicit formula for the Rasmussen invariant of odd pretzel knots

Authors: Yury Belousov, Vadim Stepaniuk
arXiv: 2610.24083
Comments: 10 pages, 7 figures
Subjects: Geometric Topology (math.GT)

We compute the Rasmussen s-invariant for all odd pretzel knots, giving an explicit closed formula. The formula extends to all slice-torus invariants that are invariant under mutation. In the absence of mutation invariance, we prove the formula under a mild condition on the pretzel parameters, which we conjecture to be unnecessary.

Official arXiv abstract page

Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds

Authors: David Gabai, Lily Qiao Li
arXiv: 2610.24190
Comments: 27 pages, 17 figures, comments welcome
Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

We show that for any closed hyperbolic 3-manifold $M$ , there are only finitely many Anosov or pseudo-Anosov flows on $M$ up to orbit equivalence.

Official arXiv abstract page

On the kissing number of hyperbolic surfaces with a small systole

Authors: S. Barré, M. Pichot
arXiv: 2610.24222
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

Parlier proved that there exists a constant $U > 0$ such that for any hyperbolic surface $S$ of genus $g$ with systole $ℓ$ the following holds: [ Kiss(S)\leq U \frac {e^\ell}\ell g ] where $K i ss (S)$ is the maximum number of homotopically distinct systoles in $S$ (Kissing number).
We show that there exists a constant $V > 0$ and a sequence $S_{n}$ of hyperbolic homogeneous surfaces of genus $g_{n} \to \infty$ with a bounded systole such that [ Kiss(S_n)\geq Vg_n ] for all $n \geq 1$ . This shows that Parlier's theorem is sharp for a fixed systole length.
Our proof uses Bruhat--Tits buildings of type $\tilde{A}_{2}$ and Hamiltonian surfaces.

Official arXiv abstract page

Analytic quantum invariants

Authors: Rinat Kashaev, Vladimir Mangazeev, Marcos Marino
arXiv: 2610.24247
Comments: 69 pages, 6 figures
Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

Using a deformed Borel-Laplace resummation, we propose a framework for repackaging quantum invariants as analytic invariants. We illustrate our approach through a detailed analysis of the coloured Jones polynomials of several simple knots, deriving explicit expressions for the resulting analytic invariants in terms of $(q, X)$ -series. These invariants are closely related to, but generally distinct from, those obtained via the conventional Borel resummation of the Melvin-Morton-Rozansky expansion. As a further example, we construct an analytic invariant from a sequence of two-variable polynomials conjectured to be the Links-Gould polynomials of the figure-eight knot, associated with the quantum superalgebras $U_{q} (gl (n ∣1))$ . Finally, we apply our technique to $q$ -series arising in the study of three-manifold invariants, revealing connections with mock modular forms.

Official arXiv abstract page

Length-convexity and domination for Hitchin representations

Authors: Subhojoy Gupta
arXiv: 2610.24606
Comments: 15 pages, 1 figure
Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Let $S$ be a closed oriented surface of genus $g \geq 2$ . For any $n \geq 3$ , let $Hit_{n} (S)$ be the Hitchin component and let $Hit_{n} (S)^{'}$ be the subset of representations that take values in $PSp_{2 m} (R)$ if $n = 2 m$ and $PSO (m, m + 1)$ if $n = 2 m + 1$ . In this note we prove that for any Hitchin representation $ρ \in Hit_{n} (S)$ there is $j \in Hit_{n} (S)^{'}$ such that $ρ$ dominates $j$ in the Hilbert length spectrum; this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for $n = 3$ . The key step is establishing a convexity of the Hilbert length function in the Bonahon-Dreyer parametrization of $Hit_{n} (S)$ ; this uses a result from the recent work of Barman-Gupta. We also prove these convexity and domination results for translation length functions and their spectra.

Official arXiv abstract page

Density of length ratios on Teichmüller space

Authors: Nhat Minh Doan, Hanh Vo
arXiv: 2610.24700
Comments: 19 pages, 2 figures
Subjects: Geometric Topology (math.GT)

We study length ratios of simple closed geodesics on finite-type hyperbolic surfaces of positive genus. We allow cusps and fix the geodesic boundary lengths. Let $d$ be the dimension of the corresponding Teichmüller space $T_{L} (S)$ . We prove that for every positive vector $(r_{0}, \dots, r_{d})$ , the set of surfaces carrying $d + 1$ distinct essential interior simple closed geodesics with lengths proportional to this vector is dense in $T_{L} (S)$ . In particular, one-holed tori with three distinct simple closed geodesics of equal length are dense. The proof uses high powers of Dehn twists, asymptotics of length differentials, and the Brouwer fixed-point theorem.

Official arXiv abstract page

Stein fillable contact structures with half Giroux torsion

Authors: Alberto Cavallo, Irena Matkovič
arXiv: 2610.24922
Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We show that there exist Stein fillable contact 3-manifolds which contain a separating half Giroux torsion layer. Our examples are on Seifert fibred spaces over the sphere.

Official arXiv abstract page

A counterexample to the geometric triangulation conjecture of Costantino, Frigerio, Martelli and Petronio

Authors: Ke Feng, Aijin Lin
arXiv: 2610.25217
Comments: 53 pages, 5 figures, 6 tables. Comments are welcome!
Subjects: Geometric Topology (math.GT)

Costantino, Frigerio, Martelli and Petronio conjectured that an ideal triangulation in which every edge has valence at least six can be realized by hyperbolic partially truncated tetrahedra on the compact manifold it defines. We show that this is false. Starting from the figure-eight knot complement, we construct a triangulation with $2688$ tetrahedra and edge valences six and eighteen whose manifold is hyperbolic with totally geodesic boundary, but which admits no such realization. The obstruction is a critical point of the covolume energy of Luo and Yang at which three tetrahedra have collapsed: convexity then excludes every geometric competitor, an Euler characteristic count excludes edges of length zero, and the critical point itself is located by a finite certificate in exact integer arithmetic. The obstruction persists in every finite cover, while a complementary theorem shows that no counterexample can have only two edge lengths or only two orbits of edges.

Official arXiv abstract page

Negative-Definite Plumbings and New Unknotting Numbers

Authors: Seong-Jin Lee
arXiv: 2610.25430
Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Algebraic Topology (math.AT)

We use Neumann's normal form to decide whether the double branched cover of a knot bounds a negative-definite plumbing tree in either orientation, starting from the recognised Seifert or graph-manifold structure of the cover. For many non-alternating knots with at most thirteen crossings whose unknotting numbers were unknown, this identifies the cases in which the Heegaard Floer homology of the cover can be computed from such a plumbing, and shows for most of the others that it cannot. We determine 112 new unknotting numbers. For 43 of them, the lower bound rests on Heegaard Floer homology computed from the plumbing; for the other 69, it comes from linking pairings and from a theorem of McCoy and Park on special alternating knots. Finally, we show that the knot 13n1587 is a further explicit counterexample to the Bernhard--Jablan conjecture, after 13n3370.

Official arXiv abstract page

Biquandle Image Quivers

Authors: Geunyoung Kim, Sam Nelson, Puttipong Pongtanapaisan
arXiv: 2610.25593
Comments: 11 pages
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We introduce a new family of weighted quiver-valued invariants of oriented classical and virtual knots and link, surface-knots and surface-links, and immersed surface-knots and surface-links associated to finite biquandles. We derive several polynomial invariants from these quivers via decategorification. We provide examples to illustrate the computation of the invariant as well as toy examples showing tables of invariant values for small biquandles.

Official arXiv abstract page

Which Immersions into Euclidean Space are Boundary Immersions?

Authors: Austin Konkel
arXiv: 2610.25618
Comments: 27 pages, 16 figures. This paper here is a condensed portion of the author's PhD thesis in progress. A.I. tools were used only to help create figures and to survey for existing literature. Comments are encouraged!
Subjects: Geometric Topology (math.GT)

Here we study the problem of whether a given codimension one immersion $f$ of a closed and connected manifold into Euclidean space can be extended to an equidimensional immersion $F$ of some compact manifold $M$ whose restriction to the boundary parameterizes the image $S$ of $f$ . Such an extension $F$ exists if and only if (i) simple conditions on orientations and wrapping numbers are satisfied and (ii) a distinguished finite subset of the first cohomology of $S$ with coefficients in a certain sheaf of finite groups contains the preferred point.

Official arXiv abstract page

Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Authors: Atsuhide Mori
arXiv: 2610.25878
Comments: 15 pages. Companion to arXiv:2609.30406
Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Symplectic Geometry (math.SG); Quantum Physics (quant-ph)

This paper is a non-planar counterpart of the author's earlier work on planar contact structures and topological quantum computation. There the page is planar and all branch points are paired by disjoint liftable arcs, compatibly with the joint eigenlines of the Ising representation and with Calabi-Yau (CY) Stein fillings, those with vanishing first Chern class. Here the page has genus g and connected binding, presented as a simple degree-three cover of the disk. Every maximal system of disjoint liftable arcs has exactly g members and leaves two branch points unused. The half-twist interchanging the two points is not liftable, but its cube is, and lifts to the identity. An integral residue criterion supplies the CY test. A factorization localized on a contact connected sum is automatically CY. Unlike in the planar case, a CY factorization need not be the shortest. Once the fusion channels of the g pairs are fixed, Ising fusion determines the channel of the remaining pair, and braiding acts there only by scalars. Fibonacci fusion can leave that channel undetermined, and the powers of the cube then act by ten distinct relative phases on a single joint eigenspace, while the CY Stein filling stays fixed

Official arXiv abstract page

Topological mapping class groups of non-simply connected 4-manifolds

Authors: Jae Choon Cha
arXiv: 2610.26400
Comments: 37 pages, 4 figures
Subjects: Geometric Topology (math.GT)

We construct topological isotopy invariants of homeomorphisms of non-simply connected 4-manifolds using the topological Dax invariant of disks and the Freedman-Quinn invariant of spheres. Our invariants are effective for various non-smoothable 4-manifolds without assuming that the fundamental groups are good. They detect infinite-rank free abelian direct factors of the pseudoisotopy-trivial and homotopy-trivial mapping class groups, and give lower bounds on the Prüfer rank of the Torelli groups modulo homotopy-trivial mapping classes, including examples of infinite Prüfer rank. Our method is also useful for smooth 4-manifolds. We construct stably smoothable homeomorphisms of the Enriques surface and many Dolgachev surfaces that act trivially on homology but are not homotopic to the identity.

Official arXiv abstract page

The Structure of the Ghost and Swapping Algebras

Authors: Caleb Ashley, Ming Hong Tee
arXiv: 2610.26726
Subjects: Geometric Topology (math.GT)

The swapping algebras and the ghost algebras were introduced in the papers of Labourie and Bridgeman--Labourie to encode the Poisson structure on spaces of length functions and generalized cross-ratio functions associated to $Θ$ -Anosov representations. They used these algebras to show that correlation functions are stable under the Poisson bracket associated to Atiyah-Bott-Goldman's symplectic form and that length functions are convex along the Hamiltonian flow of other length functions. In this paper, we study the algebraic relationship between these two algebras. In particular, we show that the kernel of a natural surjection from the ghost algebra onto the extended swapping algebra is generated by five families of relations. Consequently, we obtain an isomorphism from an explicit quotient of the ghost algebra onto the extended swapping algebra.

Official arXiv abstract page

Uniform convergence of canonical discrete uniformization on equilateral approximations

Authors: Aijin Lin, Chunlei Liu
arXiv: 2610.26798
Comments: 50 pages
Subjects: Geometric Topology (math.GT)

We prove uniform convergence of the canonical discrete uniformization maps for every sequence of equilateral polygonal disks whose mesh tends to zero and whose marked Jordan boundaries converge uniformly. The limit is the normalized Riemann map, with convergence on the entire closed reference triangle. This proves Conjecture 7.3 of Luo, Sun and Wu for the original construction with Delaunay flips. The proof combines closed-domain compactness, marked lattice limits, a packing and resistance estimate for logarithmic cusp scales, and an envelope comparison principle, followed by a local affine energy argument.

Official arXiv abstract page

Cross submissions

Asymptotics for the number of saddle connections on a slit torus

Authors: David Aulicino, Howard Masur
arXiv: 2610.24429
Cross-list: math.DS
Comments: 16 pages, 2 figures
Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

We consider a flat torus with a marked line segment $β$ and prove linear asymptotics for closed curves on the torus disjoint from the interior of the line segment. We also consider segments that begin and terminate at the end points of $β$ whose interiors are disjoint from $β$ and prove $L lo g L$ asymptotics. In all cases, after quotienting by the growth rate, we compute the constant explicitly, which differs depending on whether or not $β$ lies in a periodic direction on the torus. In particular, when $β$ lies along a non-periodic trajectory, we show the constant is $2/ (∣ β ∣ ζ (2))$ . Moreover, we prove that as a function of the holonomy vector of $β$ , the constants are continuous at the points where $β$ lies in a non-periodic direction.

Official arXiv abstract page

The minimum number of vertices of a triangulation of $RP^{5}$

Authors: Xinrui Chang (Fudan University)
arXiv: 2610.25171
Cross-list: math.CO
Comments: 9 pages; ancillary files include verification code and covering certificates
Subjects: Combinatorics (math.CO); Algebraic Topology (math.AT); Geometric Topology (math.GT)

We give a computer-assisted proof that every finite simplicial triangulation of real projective five-space has at least twenty-four vertices. Together with the construction of Guyer, Steinerberger, and Yang, this determines the minimum. A cup-product argument produces an induced subcomplex on sixteen or seventeen vertices. Transfer to the spherical double cover then imposes a covering condition on its lifted faces. In the sixteen-vertex case, a direct capacity bound excludes twenty-two vertices. For twenty-three vertices, a cochain extension argument reduces every relevant completion to one of 115 critical signed complexes; integer weights and exact covering certificates exclude all of them. The argument requires no assumption that the triangulation is combinatorial.

Official arXiv abstract page

New submissions

Explicit formula for the Rasmussen invariant of odd pretzel knots

Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds

On the kissing number of hyperbolic surfaces with a small systole

Analytic quantum invariants

Length-convexity and domination for Hitchin representations

Density of length ratios on Teichmüller space

Stein fillable contact structures with half Giroux torsion

A counterexample to the geometric triangulation conjecture of Costantino, Frigerio, Martelli and Petronio

Negative-Definite Plumbings and New Unknotting Numbers

Biquandle Image Quivers

Which Immersions into Euclidean Space are Boundary Immersions?

Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Topological mapping class groups of non-simply connected 4-manifolds

The Structure of the Ghost and Swapping Algebras

Uniform convergence of canonical discrete uniformization on equilateral approximations

Cross submissions

Asymptotics for the number of saddle connections on a slit torus

The minimum number of vertices of a triangulation of

Related content

New submissions

Explicit formula for the Rasmussen invariant of odd pretzel knots

Finiteness of Anosov and pseudo-Anosov flows on hyperbolic 3-manifolds

On the kissing number of hyperbolic surfaces with a small systole

Analytic quantum invariants

Length-convexity and domination for Hitchin representations

Density of length ratios on Teichmüller space

Stein fillable contact structures with half Giroux torsion

A counterexample to the geometric triangulation conjecture of Costantino, Frigerio, Martelli and Petronio

Negative-Definite Plumbings and New Unknotting Numbers

Biquandle Image Quivers

Which Immersions into Euclidean Space are Boundary Immersions?

Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Topological mapping class groups of non-simply connected 4-manifolds

The Structure of the Ghost and Swapping Algebras

Uniform convergence of canonical discrete uniformization on equilateral approximations

Cross submissions

Asymptotics for the number of saddle connections on a slit torus

The minimum number of vertices of a triangulation of RP5

Related content

The minimum number of vertices of a triangulation of $RP^{5}$