math.GT daily digest: 11 submissions for 16 October 2026

New submissions

Crystallization of the undertile

• Authors: Arnaud Chéritat
• arXiv: 2610.20663
• Comments: 45 pages, 81 figures
• Subjects: Geometric Topology (math.GT); Materials Science (cond-mat.mtrl-sci); Combinatorics (math.CO)

We classify all periodic tilings with a monotile known as $R1$ and called here the undertile, which underlies several aperiodic monotiles such as the Taylor-Socolar one. The classification is done using deflation, through a study of fracture lines between zones where deflation can be performed. A full classification of fracture patterns is given, that also applies to non-periodic tilings with the undertile. This can be used to reprove non-periodicity of some existing monotiles based on the undertile.

A counterexample to a conjecture of Plotnick on knot exteriors

• Authors: Tommy Hofmann, John Nicholson
• arXiv: 2610.21051
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Number Theory (math.NT)

We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.

Exotic diffeomorphisms on closed aspherical $4$-manifolds

• Authors: Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, Juan Muñoz-Echániz
• arXiv: 2610.21083
• Comments: 18 pages. Comments are welcome!
• Subjects: Geometric Topology (math.GT)

We study a class of diffeomorphisms of closed manifolds obtained from Davis' reflection trick construction, which are obtained by reflecting a diffeomorphism supported in the fundamental chamber of such a manifold. We use this construction to exhibit the first examples of closed aspherical $4$-manifolds with diffeomorphisms that are topologically isotopic to the identity map, but not smoothly isotopic to it.

Random Quotients of Hyperbolic Spaces with Respect to WPD Actions and Linear Connectivity of the Boundary

• Authors: Inhyeok Choi, Henry Talbott
• arXiv: 2610.21141
• Comments: 77 pages, 13 figures
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We study random quotients of acylindrically hyperbolic groups and hierchically hyperbolic groups, whose (partial) hyperbolicity is captured by their actions on Gromov hyperbolic spaces. We show that if an acylindrically hyperbolic group $G$ acts on a Gromov hyperbolic space $X$ coarsely transitively with a WPD element, then the quotient of $X$ by the normal closure of a random element of $G$ has linearly connected boundary. If $G$ is a group with a (relative) hierarchically hyperbolic group structure, with a quasi-tree top domain $X$, then in fact the quotient of $X$ by the normal closure of a random element of $G$ has boundary homeomorphic to the Nöbeling curve.

Holomorphic Triangles in the Heegaard Floer Pillowcase

• Authors: Eaman Eftekhary, Marzieh Souri
• arXiv: 2610.21153
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

The Heegaard Floer pillowcase is a recurring local model for rational tangle replacement and exact-triangle constructions. We compute the triangle product for its standard rational-slope triples of curves and encode all contributing classes in a finite generating polynomial. One slope ordering gives a vanishing map. For the other, a closed form for the generating function is computed. One may assume that the slopes are and . When an integer lies between the two finite slopes, the closed form has a particularly simple description and there are exactly contributing classes. We also compute the absolute Maslov bi-gradings of the intersection generators in the employed model, relating the planar labels to the graded Floer groups of two-bridge knots and links.

Scalar curvature, volume, and Weil-Petersson translation length

• Authors: Cameron Gates Rudd
• arXiv: 2610.21885
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We construct metrics on pseudoAnosov mapping tori that make every fiber an intrinsically hyperbolic stable minimal surface, and which have scalar curvature identically -6. Using this, we give a new proof (with improved constant) of an inequality bounding the hyperbolic volume of a pseudoAnosov mapping torus by the Weil-Petersson translation length (and consequently, the entropy) of its monodromy.

$h$-cobordisms between $1$-connected manifolds with boundary

• Authors: Ansel Goh
• arXiv: 2610.22305
• Comments: 25 pages
• Subjects: Geometric Topology (math.GT)

We classify $h$-cobordisms between compact simply-connected $4$-manifolds with boundary in both the smooth and topological categories. This extends a result of Kreck, which proved the corresponding result for closed manifolds. As an application, we recover the injectivity portion of Orson-Powell's computation of the mapping class group for simply-connected topological $4$-manifolds with boundary.

Cross submissions

Resolutions of orbifold generalised triangle groups

• Authors: Sean P. Carroll, Thomas Csizmadia, Jeroen Schillewaert
• arXiv: 2610.20932
• Cross-list: math.GR
• Comments: 24 pages, 3 figures
• Subjects: Group Theory (math.GR); Algebraic Topology (math.AT); Geometric Topology (math.GT)

Let be a generalised triangle group in which $W$ is the Farey word of a rational slope , and suppose that $T$ is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let be the associated central extension. We compute an explicit free resolution of the trivial module over $\mathbb{Z}G$ of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over $\mathbb{Z}T$ which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family in 1983. Since then only the family has been treated, by Ellis and Williams, and for that family our resolution over $\mathbb{Z}T$ has strictly smaller rank in every degree.

Quantum Color Newton Rings

• Authors: Honghuai Fang
• arXiv: 2610.21102
• Cross-list: math.QA
• Comments: Comments are welcome!
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)

We study integral Newton rings attached to symmetric quantum colors. We show that their boundary values at a deleted node are exactly the elements of Habiro's cyclotomic completion, and we describe their quantum neighborhoods at uniformly in the rank. The symmetric-color invariants of a knot are the values of an element of such a ring. In its expansion at , the coefficient of has denominator dividing the th power of the Alexander polynomial, with integral numerator, and this exponent is sharp. For and at , the period matrices of the rank-two coefficient recurrences have spectral modules of exact order three in the Picard group of an affine elliptic curve.

Augmentation curve for torus knots

• Authors: Alexander Hock, Sergey Shadrin
• arXiv: 2610.21585
• Cross-list: math-ph
• Comments: 27 pages
• Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime $(r,s)$. Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic--Vafa conjecture for all torus knot augmentation curves.

Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$

• Authors: Christian El Emam, Filippo Mazzoli, Nicholas Rungi, Andrea Seppi, Andrea Tamburelli
• arXiv: 2610.22288
• Cross-list: math.DG
• Comments: 68 pages
• Subjects: Differential Geometry (math.DG); Complex Variables (math.CV); Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct a deformation space for minimal immersions into $\mathrm{PSL}(2,\mathbb{C})$ with its holomorphic Killing form, and define an explicit, mapping class group invariant, complex hyperKähler structure on such deformation space. Such complex hyperKähler structure restricts to: Donaldson's hyperKähler structure on a totally real copy of almost-Fuchsian space, Mazzoli-Seppi-Tamburelli's para-hyperKähler structure on a totally real copy of maximally globally hyperbolic manifolds, and Loustau-Sanders' complex bi-Kähler structure on a holomorphic copy of quasi-Fuchsian space. Moreover, all these loci are totally geodesic. We discuss other geometric aspects of this space, such as the Hamiltonian property of a natural -action, involving the Goldman symplectic form and a complex-valued area functional.