math.GT daily digest: 20 submissions for 27 October 2026

New submissions

On the minimality of the compact right-angled 120-cell

• Authors: Lizi Guo, Fangting Zheng
• arXiv: 2610.33053
• Comments: 34 pages, 9 figures, 8 tables. Programs and data are available at this https URL
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Metric Geometry (math.MG)

We prove that a compact right-angled polytope in hyperbolic 4-space has at least 120 facets, with equality only for the right-angled 120-cell. This answers a question of Potyagailo and Vinberg. As a consequence, the right-angled 120-cell is the unique compact right-angled 4-polytope of minimal volume and f-vector.

On the Upsilon invariant and the Baldwin-Vela-Vick b-invariant of fibered knots

• Authors: Linh Truong
• arXiv: 2610.33280
• Comments: 12 pages, 1 figure
• Subjects: Geometric Topology (math.GT)

For a null-homologous fibered knot $K$ in a rational homology sphere, Baldwin and Vela-Vick define a refinement $b(K)$ of the Ozsváth-Szabó contact invariant. We prove that if the Ozsváth-Stipsicz-Szabó concordance invariant Upsilon has slope for some , where $g(K)$ is the genus of $K$, then . This extends a result of He, Hubbard, and the author on the Upsilon invariant and right-veering monodromies. We also prove that the knot Floer homology of a fibered knot in the Alexander grading is nonvanishing. As an application, we give many families of fibered cable knots whose $b(K)$ invariant grows arbitrarily large. Finally, we prove that if a monodromy of a fibered knot is neither right-veering nor left-veering, then the third-to-top term of knot Floer homology is nonvanishing.

Delambre--Gauss Formulas in Complex Hyperbolic Space

• Authors: Jiming Ma, Baohua Xie
• arXiv: 2610.33468
• Comments: 22 pages, 2 figures
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We establish a Delambre--Gauss matrix identity for oriented right-angled hexagons in the complex hyperbolic plane. The six complex length parameters of a hexagon determine two alternating matrices whose product is central in (\mathrm{SU}(2,1)). The resulting nine entry-wise equations extend the Delambre-Gauss formula of Tan, Wong, and Zhang in 4-dimensional hyperbolic geometry to this complex hyperbolic setting.

Dancing circles

• Authors: Danica Kosanović
• arXiv: 2610.33867
• Comments: 26 pages, 9 figures. To appear in the Proceedings of the 2025 Georgia International Topology Conference
• Subjects: Geometric Topology (math.GT)

Recent results on diffeomorphisms of $4$-manifolds make essential use of families of embeddings of $1$-manifolds into $4$-manifolds. We survey what is currently known about such families.

The geometric combinatorics of $k$-shuffle polynomials

• Authors: Chloé Postel-Vinay
• arXiv: 2610.33890
• Comments: 47 pages, 37 figures. This paper is the second one in a series of two. The first paper is available at arXiv:2610.28986
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Complex Variables (math.CV); Group Theory (math.GR)

This paper is the second in a series of two which aims at introducing a new family of groups called $k$-shuffle braid groups, denoted . We focus here on the geometrical origins of these groups, as first observed by Calegari in arXiv:2105.11265. The spaces we are interested in are spaces of complex monic polynomials of degree $n$ whose critical values lie in . We show that the $k$-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of $k$ translates of the discriminant variety. Moreover, we show that the $k$-shuffle braid groups inject into the larger braid groups .

Contact 3-manifolds with infinitely many rational homology ball Stein fillings

• Authors: Burak Ozbagci
• arXiv: 2610.34381
• Comments: 22 pages, 13 figures
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct the first examples of contact $3$-manifolds admitting infinitely many pairwise non-homeomorphic Stein fillings which are rational homology balls. Our examples are planar. They are obtained by a simple modification of Baykur's recent planar contact $3$-manifold $(Y,\xi )$ with infinitely many pairwise non-homeomorphic Stein fillings, and the fillings are distinguished by their fundamental groups. For one of these rational homology balls we give an explicit Kirby diagram, with a single $1$-handle and a single $2$-handle attached along a trefoil, which exhibits its boundary as $(0,-8)$-surgery on a two-component link. We also identify the smooth topology of Baykur's example: $Y$ is obtained by $(0,-8)$-surgery on the link , and it is Seifert fibered over the Klein bottle with one exceptional fiber. Finally, we construct further Stein fillings of $(Y,\xi )$, which realize all the groups , , as their fundamental groups.

On Khovanov homology of virtual links

• Authors: Igor Nikonov
• arXiv: 2610.34438
• Comments: 14 pages
• Subjects: Geometric Topology (math.GT)

We show that the Khovanov homology of virtual knots defined by V.O. Manturov and the Khovanov homology defined by D. Tubbenhauer coincide.

Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres

• Authors: Xiaojun Lai, Xueyuan Wan
• arXiv: 2610.34743
• Subjects: Geometric Topology (math.GT)

We study the twelve-vertex flag triangulation of and the Davis $4$-manifold associated with it. We prove that is, up to isomorphism, the unique flag triangulation of with twelve vertices in which every edge is contained in a square; this proves Milizia's Conjecture~6.31. We also prove a general criterion for the tangent Euler class of a Davis $4$-manifold to be bounded, and we verify it for by means of an explicit symmetric form. Since the Euler class takes the value $256$ on the fundamental class of , the duality between bounded cohomology and simplicial volume gives . This answers Milizia's Question~1.7.

Floer lasagna modules for some 2-handlebodies

• Authors: Akram Alishahi, Eugene Gorsky, Beibei Liu
• arXiv: 2610.34744
• Comments: 52 pages
• Subjects: Geometric Topology (math.GT)

Morrison, Walker and Wedrich used Khovanov homology to define the skein lasagna module for a 4-manifold with a link in its boundary. The skein lasagna module was shown to be a deep and powerful invariant; in particular, Ren and Willis proved that it detects exotic smooth structures. Still, computing the skein lasagna module outside some carefully chosen grading range remains an important open problem. Motivated by this construction, Chen defined the Floer lasagna module using Heegaard Floer link homology. In this paper, we give a complete, but partially conjectural description of Floer lasagna modules for a class of 4-manifolds obtained by adding 2-handles along plumbed L-space knots (with certain framing constraints). In particular, we prove that for it is infinite-dimensional, and conjecture that for it is infinite-dimensional as well. We compute this module for (and conjecturally for ) in all Alexander and Maslov gradings. This strengthens and extends the recent results of McCulloch.

The Alexander polynomial of twisted torus knots in the complementary range

• Authors: Adnan, Kyungbae Park
• arXiv: 2610.34813
• Comments: 21 pages, 4 figures. Comments and suggestions are welcome
• Subjects: Geometric Topology (math.GT)

A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1> A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}> A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}<r<=|p|+|q| by placing the knot on a standard genus-two Heegaard surface and applying Fox calculus. We then combine the two calculations into a unified Alexander-polynomial formula valid for 1> A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}<r<=|p|+|q| by placing the knot on a standard genus-two Heegaard surface and applying Fox calculus. We then combine the two calculations into a unified Alexander-polynomial formula valid for 1<r<=|p|+|q|. In the complementary range, we prove monicity and give an explicit formula for the breadth. This determines the Seifert genus for nonnegative twists and gives a genus lower bound for negative twists. We also exhibit infinite two-parameter families of non-L-space knots in the complementary range.

A lifting defect for oriented $t_3$ and $\bar t_6$ moves

• Authors: Wenbiao Liu
• arXiv: 2610.34831
• Comments: 24 pages, 4 figures
• Subjects: Geometric Topology (math.GT)

We study oriented - and -moves through the problem of lifting Fox $3$-colorings to colorings by a nine-element Alexander quandle. We define a nonnegative integer lifting defect $\delta$ for oriented links and prove that it is invariant under these moves. For knots, the defect equals the rank of an alternating bilinear form obtained from a Seifert matrix, so it is even. We also prove that it is additive under connected sum. Let be the connected sum of copies of . Then , so these knots represent infinitely many distinct equivalence classes under the oriented moves, and none is equivalent to an unlink. After forgetting orientation, each can be transformed into an unlink with components by $3s$ ordinary $3$-moves.

On a family of stably handle-slide trivial R-links

• Authors: Wenjie Diao
• arXiv: 2610.35120
• Comments: 23 pages, 22 figures, comments welcome
• Subjects: Geometric Topology (math.GT)

An R-link is an n-component link L in S^3 such that Dehn surgery on L yields #^nS1 \times S2. Adapting work of Gompf, Scharlemann and Thompson, Meier and Zupan produced a family of R-links L(p, q; c/d), where the pairs (p, q) and (c, d) are relatively prime and c is even. For most of link in this family, whether it is handle-slide trivial remains open. In this work, we show one subfamily, L(3, 2; 4/d), are stably handle-slide trivial for all d.

Knot Floer complexes of generalized L-space knots

• Authors: Shengyu Zou
• arXiv: 2610.35305
• Comments: 89 pages, 122 figures, 26 tables. Most figures depict subquotient complexes by displaying their generators and differentials
• Subjects: Geometric Topology (math.GT)

In this paper, we study height 4 knots, which generalize L-space knots and almost L-space knots studied by Ozsváth--Szabó, Baldwin--Sivek, and Binns. A knot is called a height $m$ knot if for all integers , where is the hat version of Heegaard Floer homology introduced by Ozsváth--Szabó. In particular, height 0 and height 2 knots are exactly L-space knots and almost L-space knots, respectively. We prove that the knot Floer complex of a height 4 knot $K$ admits a basis that is horizontally and vertically simplified--an hv-basis for short. Moreover, we show that under such an hv-basis, the horizontal and vertical differentials fall into six cases, generalizing the results for L-space knots and almost L-space knots. The hv-basis of the knot Floer complex of $K$ can determine, via an explicit algorithm, the immersed curve of Hanselman-Rasmussen-Watson for the knot complement of $K$. Consequently, our results facilitate further exploration of computations of for the gluing of two height 4 knot complements.

Classifying Leibniz racks

• Authors: Mohamed Elhamdadi, Bryce Virgin
• arXiv: 2610.35534
• Comments: 29 pages. Comments are welcome
• Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Category Theory (math.CT); Differential Geometry (math.DG)

Lie quandles were defined by T. Fritz as a nonlinear generalization of Lie algebras, to serve as a nonlinear algebras of observables in generalized form of Hamiltonian mechanics. These objects were subsequently generalized in a prequel article to objects referred to as smooth Lie group families of smooth racks by the authors. In this article, we investigate a particular system of nonlinear functional differential equations whose solution in general provides a pathway to the classification of smooth Lie group families of smooth rack structures on any given smooth manifold. We then discuss a particularly practical ansatz for solving these equations, and use it to construct new families of examples, as well as prove a result on the sense in which the Bloch sphere is the unique Lie quandle structure on the sphere.

Cross submissions

The Number of Polytope Trees and Beyond

• Authors: Ali Aliev, Natalia Amburg
• arXiv: 2610.33344
• Cross-list: math.CO
• Comments: 54 pages, 13 figures
• Subjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Geometric Topology (math.GT)

We study the symmetry-weighted enumeration of complexes formed by gluing copies of a regular polytope. For polytope trees and complexes obtained by attaching such trees to a fixed initial complex, we derive cut-and-join recurrences, differential equations for generating functions, explicit counting formulas, and asymptotic estimates. We then extend the method to complexes with cycles in their dual graphs, enumerating them by the numbers of polytopes and free facets. We derive recurrence relations and partial differential equations for the corresponding bivariate generating functions. In this setting, we obtain recurrences for connected configurations and explicit coefficient formulas for possibly disconnected configurations, both with and without a distinguished initial complex.

Volume of area-bounded PGL3-Hitchin moduli space

• Authors: Qiongling Li, Beibei Liu, Weixu Su, Zhe Sun
• arXiv: 2610.33417
• Cross-list: math.DG
• Comments: 25 pages, 7 figures. This preprint supersedes the earlier preprint arXiv:2001.01295
• Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Geometric Topology (math.GT)

For the -Hitchin moduli space of unmarked convex structures on the surface with fixed boundary data $\mathbf{L}$ when , we investigate the subset whose canonical Hilbert--Busemann area is bounded above by $t$. We show that the Goldman symplectic volume of this subset is at most , where $\lambda$ is the largest total-root boundary length for and is set to $0$ for $m=0$. As a consequence, We obtain the explicit upper bound for the -weighted volume.

Aspherical Symplectic Manifolds with Positive First Chern Degree

• Authors: Jingcao Wu
• arXiv: 2610.34998
• Cross-list: math.SG
• Comments: 10 pages, comments are welcome!
• Subjects: Symplectic Geometry (math.SG); Differential Geometry (math.DG); Geometric Topology (math.GT)

For every integer , we construct a closed aspherical symplectic manifold of real dimension $2n$ with positive first Chern degree. Each example is a flat torus bundle over the same compact hyperbolic surface, with arithmetic monodromy arising from an explicit indefinite quaternion division algebra over $\mathbb{Q}$. We identify its symplectic first Chern class with the pullback of the canonical class of the base and compute its degree explicitly. Asphericity rules out nonconstant pseudoholomorphic spheres for every tame almost complex structure. Thus these examples give a negative answer to the symplectic uniruledness question of Boucksom--Demailly--Păun--Peternell in every even real dimension at least ten.

Non-discrete complex hyperbolic -triangle groups

• Authors: Wei Liao, John R Parker
• arXiv: 2610.35197
• Cross-list: math.GR
• Comments: 19 pages
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

In this paper, we study complex hyperbolic triangle groups of type with . We show that if the element is elliptic, then the group is either non-discrete or it belongs to a short list of known discrete groups.

The flipping principle for boundedly acyclic covers and colorings

• Authors: Pietro Capovilla, Alberto Casali, Marco Moraschini
• arXiv: 2610.35222
• Cross-list: math.AT
• Comments: 28 pages
• Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)

Based on the theory of multicomplexes and other ideas by Gromov, we describe the flipping principle. This is an easy geometric criterion that forces the vanishing of the Gromov norm of homology classes. We use it to provide proofs based on a common geometric principle of the Vanishing Theorem and the Amenable Reduction Lemma by Gromov in the case of amenable and boundedly acyclic covers and colorings. We also strengthen the conclusion of the Amenable Reduction Lemma by showing that it implies the -invisibility of the homology classes and not only the vanishing of their Gromov norm.

The Weyl Problem for Planar CBB$(-1)$ Surfaces in Hyperbolic $3$-Space

• Authors: Xinrong Zhao
• arXiv: 2610.35744
• Cross-list: math.DG
• Comments: 58 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT); Metric Geometry (math.MG)

In this paper, we study the Weyl problem for noncompact convex surfaces in hyperbolic $3$-space . We prove that any planar CBB$(-1)$ surface with countably many ends can be isometrically embedded as the boundary of a closed convex subset such that each connected component of the ideal boundary of $X$ is either a round disk or a single point. By known rigidity results, such an embedding is unique up to isometries of . Moreover, we show that the assumption of countably many ends can be dropped if Koebe's circle domain conjecture holds.