math.GT daily digest: 15 submissions for 30 October 2026

New submissions

A short proof of the universality of the fundamental shadow links

• Authors: Yuxuan Yang
• arXiv: 2610.37718
• Subjects: Geometric Topology (math.GT)

We give a constructive three-dimensional proof of Costantino and Thurston's universality theorem for fundamental shadow links. More precisely, every compact connected oriented three-manifold with empty or toroidal boundary can be obtained by integral Dehn filling along a subset of the boundary components of a fundamental shadow link exterior.

Loop Space Models and Integral Homology of Connected Quandles

• Authors: Takefumi Nosaka
• arXiv: 2610.38066
• Comments: 33 pages, 1 figure
• Subjects: Geometric Topology (math.GT)

We construct a marked integral delooping of a connected quandle's covering monoid, realizing its rack space as a homotopy fiber. For finite connected quandles at primes $p$ not dividing a point-stabilizer order, we determine the local type and the ordinary rack cohomology shuffle algebra over fields of characteristic $p$. We compute integral quandle homology in all positive degrees for finite connected regular cyclic Alexander quandles and the tetrahedral quandle. At such primes dividing the quandle order, rack-space homotopy groups have exponentially many cyclic direct summands of fixed $p$-power order in each sufficiently large degree.

Sharp dimension bounds for right-angled hyperbolic polytopes

• Authors: Alexander Kolpakov
• arXiv: 2610.38900
• Comments: 6 pages; GitHub repository at this https URL
• Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

We determine the largest dimensions in which right-angled hyperbolic polytopes of finite volume can exist. The dimension is at most eight in general, and at most four when all vertices lie at infinity. Both bounds are attained, improving the previously known upper bounds of twelve and six. Our proofs use elementary properties of hyperbolic polytopes together with bounds for configurations of points on spheres.

$L$-Space Knots in -like Knot Floer Homology

• Authors: Tristan Wells
• arXiv: 2610.38927
• Subjects: Geometric Topology (math.GT)

It has been of particular interest in recent years to relate categorifications of quantum knot invariants like the Jones polynomial to Floer-theoretic invariants like knot Floer homology using spectral sequences. In pursuit of Rasmussen's conjecture, Dowlin constructed a family of invariants with properties like those of Khovanov and Rozansky's -homology theories. In this paper, we explore these invariants for $L$-space knots and thin knots. In particular, for the invariants of thin knots can be computed from a change of coefficients from the minus flavor of knot Floer homology, and the same holds for an $L$-space knot exactly when an explicit condition relating $n$ to its Alexander polynomial is satisfied.

Non-concordant 2-links in simply-connected 4-manifolds

• Authors: Maggie Miller, Ryan Stees
• arXiv: 2610.38979
• Comments: 10 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

We construct an infinite family of smooth 2-component links of 2-spheres whose respective components are smoothly concordant but which are pairwise not topologically concordant in any topological $\mathbb{Z}$-homology cobordism. In particular, the are pairwise non-concordant in .

The Liouville property for geometric graphs in the plane

• Authors: Kuijun Liu, Feng Luo, Yanwen Luo, Zhenghao Rao
• arXiv: 2610.39242
• Comments: 16 pages, 4 figures
• Subjects: Geometric Topology (math.GT)

We show that every bounded discrete harmonic function with respect to the cotangent Laplacian on a Delaunay triangulation of the Euclidean plane is constant; equivalently, the cotangent-weighted triangulation has the Liouville property. Our main theorem establishes this property for a more general class of connected weighted geometric graphs in the plane. Furthermore, we show that a planar Delaunay triangulation with cotangent weights could be transient but is recurrent if edge lengths are uniformly bounded.

Higher order unknotting numbers for knots and links

• Authors: Jean-Baptiste Meilhan, Harumi Ogita, Akira Yasuhara
• arXiv: 2610.39474
• Comments: 18 pages; no AI tools were used at any stage
• Subjects: Geometric Topology (math.GT)

Ohyama, Taniyama and Yamada showed that finite type knot invariants are insensitive to the unknotting number: for any integer n and any knot K, there is an unknotting number 1 knot which cannot be distinguished from K by any finite type invariant of order $\le n$. We generalize this result in three directions. First, we prove an analogous result for all higher order unknotting number given by -moves. Second, we give several extensions to the link case. Third, we show that a similar result holds for ribbon 2-knots.

Topological Insights and Combinatorial Frameworks for Twist Knot Complements

• Authors: Biplab Basak, Mangaldeep Saha, Ayushi Trivedi
• arXiv: 2610.39593
• Comments: 23 pages, 6 figures
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

The complement of a knot in is a fundamental object in $3$-manifold topology and serves as a primary building block in the study of geometric structures on manifolds. For a large class of knots, these complements admit complete hyperbolic metrics, motivating the development of various frameworks for their classification and analysis. A twist knot is formed by applying a series of half-twists to a closed loop and linking the ends. In this article, we present explicit topological and combinatorial descriptions of all twist knot complements via the construction of suitable cell structures. Our method produces a closed $3$-dimensional pseudomanifold with a single vertex, arising from prescribed face pairings of a faceted $3$-ball. Removing the vertex yields the complement of the corresponding twist knot. The construction naturally induces ideal triangulations of these knot complements, several of which are shown to be facet-minimal. Beyond providing concrete models, our approach offers new perspectives on the topology of knot complements and leads to some open research problems.

Isospectral congruence manifolds, spectral rigidity, and a converse to Vignéras

• Authors: Justin Katz, D. B. McReynolds
• arXiv: 2610.39874
• Comments: v1. 16 pages
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Number Theory (math.NT)

Let $B$ be a quaternion division algebra over a number field $k$ that gives rise to arithmetic Fuchsian or Kleinian groups, and let $\mathcal{O}$ be a maximal order of $B$. We show that every closed orientable Riemannian manifold isospectral to a principal congruence hyperbolic surface is isometric to for some maximal order of $B$, provided that no prime factor of $\mathfrak{a}$ ramifies in $B$ and that $\mathfrak{a}$ satisfies a mild condition at the primes over $2$. This is a converse to the construction of Vignéras, which produces isospectral manifolds from maximal orders of different types. In particular the spectral genus of has at most $t(B)$ elements, and is spectrally rigid when the type number $t(B)$ is one. In dimension three we prove the same statements for manifolds that, in addition, have the same complex lengths of closed geodesics. The proof rests on a local result that characterizes, by traces alone, the subgroups of $\mathrm{SL}(2)$ over a $p$-adic ring that are conjugate into a principal congruence subgroup by traces. As applications, every principal congruence Hurwitz surface is spectrally rigid, among them Klein's quartic, the Fricke-Macbeath surface and the surfaces of the first Hurwitz triplet, which partially answers a question of Reid.

The Torelli Group and Euler Class Zero $\mathrm{PSL}(2,\mathbb{R})$ Representations

• Authors: Marit Bobb, James Farre, Peter Smillie
• arXiv: 2610.40040
• Comments: 13 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

We prove that every representation of a closed surface group of genus at least $3$ into $\mathrm{PSL}(2,\mathbb{R})$ that is sufficiently close (but not equal to) a representation that has non-compact solvable image has a simple closed separating curve that is mapped to a non-trivial elliptic element in $\mathrm{PSL}(2,\mathbb{R})$. This can be seen as evidence for an affirmative answer to a question of Bowditch and a conjecture of Goldman.

Linking-form splitting and four-dimensional genera of knots

• Authors: Seong-Jin Lee
• arXiv: 2610.40183
• Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Algebraic Topology (math.AT)

By Gilmer's splitting lemma, the linking form of a rational homology sphere that bounds a 4-manifold is the orthogonal sum of a metabolic summand and a summand presented by a form of the rank of the intersection form. The lemma underlies Gilmer's slice genus theorem and the rank criteria of Murakami-Yasuhara and Gilmer-Livingston for nonorientable surfaces and the 4-dimensional clasp number. We show that the lemma and these results, as stated, are false. For each prime we construct knots bounding smooth Möbius bands, and knots bounding smooth punctured tori with clasp number one, whose linking forms have no such summand of rank at most two. We then determine what remains true: the lemma holds up to Witt equivalence and for cube-free orders, and its conclusions about summands hold away from exceptional primes and for orientable surfaces of genus at least two. Finally, we find the optimal Witt-class bound for the topological nonorientable 4-genus.

Skein reduction and stabilization for the odd Khovanov spectra of

• Authors: Krishnendu Kar
• arXiv: 2610.40245
• Comments: 37 pages, 6 figures, 2 tables
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

For split links the even Khovanov spectrum is a smash product of the spectra of the pieces; the odd one is not. For we show that the odd spectrum is, in each quantum grading, a wedge of spheres and copies of a suspension of . Using the skein cofiber sequence at the level of spectra, we prove that for fixed the number of these copies is determined by finitely many small links and stabilizes for large $m$ and $n$. The summand of , due to Schütz and Rajapakse, persists in quantum grading for all . We conjecture that the number of copies of equals the number of copies of in the even spectrum.

Presentations for sisters of one-cusped Picard modular groups

• Authors: Jackson Young
• arXiv: 2610.40568
• Comments: 25 pages
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We generalize a method devised by Mark and Paupert for finding group presentations of the single-cusped standard Picard modular groups , where is the ring of integers of , to a more general class of lattices in $\text{PU}(2,1)$. As an application, we give presentations for the so-called \textit{sisters} of the standard Picard modular groups for and $11$. In particular, this shows that the sister of is a perfect group.

Cross submissions

Vietoris-Rips complexes of affine Coxeter groups

• Authors: Riley Decker, Matthew C. B. Zaremsky
• arXiv: 2610.38703
• Cross-list: math.GR
• Comments: 20 pages, 3 figures
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We prove that all affine Coxeter groups admit contractible Vietoris-Rips complexes. More precisely, for any affine Coxeter group there exists a finite generating set $A$ such that is contractible for all . We prove this as a special case of a more general result about certain groups. Let $G$ be a group and $H$ a subgroup. A subgroup is a supplement of $H$ in $G$ if . We prove that for with $F$ a supplement of $H$ in $G$, for generating sets and of $H$ and $G$ satisfying certain conditions involving $F$, the Vietoris-Rips complexes and are homotopy equivalent for all . In particular this applies to affine Coxeter groups, since free abelian groups admit contractible Vietoris-Rips complexes. Beyond these general results, we also investigate affine Coxeter groups specifically using the standard generating set $S$, and prove contractibility of for large enough $t$.

G-equivariant Alterfold TQFT in any dimension

• Authors: Zixuan Feng, Zhengwei Liu
• arXiv: 2610.40041
• Cross-list: math.QA
• Comments: 79 pages
• Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Category Theory (math.CT); Geometric Topology (math.GT); Operator Algebras (math.OA)

For a discrete group $G$, we construct an -dimensional $G$-equivariant Alterfold topological quantum field theory from a $G$-equivariant -algebra satisfying strong semisimplicity, complete finiteness, and Morita invariance. We use state sum on triangulation to define a partition function and prove that it is invariant under local surgery relations and is invariant under gauge transformations relative to the boundary. If the input is given a reflection-positive structure, the theory is unitary. For $n=2$ and input reconstructed from a strict faithful spherical $G$-fusion category, the restriction to ordinary $G$-cobordisms is naturally isomorphic to the Turaev--Virelizier state-sum HQFT \cite{turaev20123}. A Tambara-Yamagami example recovers the Quon language \cite{liu2017quon} as a -equivariant Alterfold TQFT.