math.GT daily digest: 12 new submissions for 10 August 2026

New submissions

Cofinal towers with vanishing homology torsion

• Authors: Qilong Guo
• arXiv: 2608.06601
• Subjects: Geometric Topology (math.GT)
• Comments: 6 Pages,no figures

Problem 3.6 in the problem list of Baykur, Kirby and Ruberman asks whether every cofinal tower \[ M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots \] of finite covers of a finite-volume hyperbolic $3$-manifold satisfies \[ \lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|} {\operatorname{vol}(M_n)} =\frac{1}{6\pi}. \] We give a negative answer. For every ideal right-angled polyhedron $P\_0$, the checkerboard manifold associated to $P\_0$ admits a cofinal tower all of whose levels are hyperbolic link complements in . Thus at every level, and the normalized logarithmic homology torsion is identically zero.

Ribbon concordance and cabling

• Authors: Jennifer Hom; JungHwan Park
• arXiv: 2608.06625
• Subjects: Geometric Topology (math.GT)
• Comments: 18 pages, 2 figures

We study ribbon concordances to cable knots. We formulate a conjecture predicting that any nontrivial knot admitting a ribbon concordance to a (p,q)-cable must itself be a (p,q)-cable. We prove the conjecture when the knot admitting the ribbon concordance is already a cable of the same companion, is a torus knot, or has genus one. We also verify it for a broad class of target cables. The proofs use a minimum-height invariant defined from the immersed-curve formulation of knot Floer homology. This invariant obstructs ribbon concordances and implies that any nontrivial knot admitting a ribbon concordance to a fibered cable knot is prime. We also establish genus bounds for knots admitting ribbon concordances to cable knots.

On two conjectures on triangulations of 2-manifolds

• Authors: John M. Campbell
• arXiv: 2608.06863
• Subjects: Geometric Topology (math.GT)
• Comments: Submitted for publication

For a closed, connected 2-manifold $M$, and for a triangulation $T$ of $M$, we write $V(T)$ in place of the vertex set associated with $T$. A cyclic coloration of a triangulation $T$ of $M$ refers to a face coloring of $T$ such that: For each , the faces incident to $v$ have distinct colors. Chen and Lawrencenko [Yokohama Math. J., 1999] conjectured that there exists a constant $C(M)$ (depending only on $M$) such that colors suffice for there to exist a cyclic coloration of a triangulation $T$ of $M$. We prove this conjecture, using a greedy algorithm related to the Euler-Poincaré formula for surface triangulations. Chen and Lawrencenko also conjectured that: If $M$ is not the projective plane and $T$ is a triangulation of $M$ that is minimal with respect to the number of vertices, then , where denotes the minimum possible number of vertices among all triangulations of the 2-manifold $M$, and where $xi(T)$ denotes the minimal value $k$ such that $T$ admits a cyclic coloration with $k$ colors. We disprove this latter conjecture via an explicit counterexample, using an 8-vertex triangulation of the Klein bottle with 16 faces. It appears that both of the Chen-Lawrencenko conjectures have remained open, prior to our work.

Computations of parabolic character schemes of knots

• Authors: Yunhi Cho; Hyuk Kim; Seonhwa Kim; Seokbeom Yoon
• arXiv: 2608.06890
• Subjects: Geometric Topology (math.GT)
• Comments: 21 pages. Some of the computational data were previously presented in arXiv:2204.00319

We compute parabolic -character schemes of knots using the parabolic quandle. To this end, we introduce sign-refined arc-colorings and show that their sign data encode the obstruction classes of the induced parabolic representations. We also establish a correspondence between the schemes defined by sign-refined arc-colorings and the parabolic character scheme. This yields a practical diagrammatic method for computing complete lists of parabolic characters, together with their multiplicities and obstruction classes. Using this method, we verify a conjecture of Bénard and Detcherry for all small knots with at most $12$ crossings.

B-index polynomial for twisted knots

• Authors: Kirandeep Kaur; Madeti Prabhakar; Vaibhav Keshari
• arXiv: 2608.06911
• Subjects: Geometric Topology (math.GT)

A twisted link is a generalization of a virtual link associated with link diagrams on closed surfaces, which may be non-orientable. In this paper, we generalize the notion of the index value for twisted knots. Based on this generalization, we introduce a polynomial invariant for twisted knots, called the -Index polynomial. Furthermore, we construct a family of twisted knots with arc shift number $n$ and determine their -Index polynomials explicitly in terms of $n$. These results demonstrate the effectiveness and sensitivity of the -Index polynomial as an invariant of twisted knots. We also study the behavior of this polynomial under mirror images and orientation reversal. Furthermore, we conclude this paper by investigating the cosmetic crossing change conjecture and establishing a condition under which a crossing does not admit cosmetic behavior.

Fox-Milnor condition for concordant knots in homology 3-spheres

• Authors: Huu-Bao Vuong
• arXiv: 2608.07039
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

This paper will show that the Alexander polynomial of a knot, which is of slice type in an oriented homology 3-sphere, obeys the Fox-Milnor polynomial condition. A relation between Alexander polynomial of concordant knots in an oriented homology 3-sphere is established.

Involutive Khovanov homology and equivariant knots II

• Authors: Taketo Sano
• arXiv: 2608.07114
• Subjects: Geometric Topology (math.GT)
• Comments: 49 pages. Comments are welcome!

In the spirit of Bar-Natan's formulation of Khovanov homology for tangles, we extend the framework of involutive Khovanov homology to involutive tangles. This enables a divide-and-conquer computation of involutive Khovanov homology and the equivariant Rasmussen invariant, which results in a significant speedup for the algorithmic computation. With this, we obtain new examples of strongly invertible knots for which no slice disk is smoothly isotopic rel boundary to its symmetric counterpart. In particular, we show that the Whitehead doubles of the pretzel knots and admit exotic pairs of slice disks.

Reachability under arc crossing changes

• Authors: Bo Chen; Jinbo Geng; Zerui Wu
• arXiv: 2608.07319
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

Cericola proved that every knot diagram can be transformed into an ascending unknot diagram by arc crossing changes. We refine this result for all diagrams on a fixed R1-reduced classical knot shadow with more than three crossings. Apart from at most two source diagrams, any two diagrams of the same state parity are mutually reachable. Each source diagram reaches every non-source diagram of its parity but cannot be reached from another diagram. A local five-occurrence condition on the marked Gauss word characterizes the sources and thereby determines the full directed reachability relation.

Profinite rigidity in lattices of

• Authors: Xiaoyu Xu
• arXiv: 2608.07350
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 96 pages, 5 figures; comments are welcome!

A finitely generated group is profinitely rigid among a class of finitely generated groups if it can be distinguished among this class by its set of finite quotient groups. This paper proves that all lattices in are profinitely rigid among themselves. In addition, for any lattice , it is proven that , where denotes the profinite completion of $Gamma$.

Cross submissions

Holomorphic and Formal First Integrals for Foliations of Codimension One on Complex Analytic Space Germs

• Cross-list: math.CV
• Authors: Victor León; Bruno Scárdua
• arXiv: 2608.06491
• Subjects: Complex Variables (math.CV); Geometric Topology (math.GT)

We study holomorphic and formal first integrals for germs of codimension-one holomorphic foliations on normal complex analytic spaces. In dimension two, under the assumption that the dual graph of the exceptional divisor of a resolution is a tree, we prove that the foliation admits a holomorphic first integral if and only if its leaves are closed outside the singular point and only finitely many leaves accumulate at that point. This extends a classical integrability theorem of Mattei and Moussu to singular ambient spaces. We also prove a holomorphic prolongation theorem for normal quotient germs admitting a smooth quasi-étale cover and a smooth connected lift of a generic two-dimensional section. We record, in addition, a conditional formal prolongation statement under depth assumptions on the conormal powers and an injectivity condition for the corresponding differential-form obstruction modules. Under the quotient-prolongation hypothesis, and with a reduced tangent cone where formal restriction must be detected, the higher-dimensional integrability results follow from their surface counterparts. We give a reduced nonnormal example satisfying both dynamical conditions but admitting no holomorphic first integral, showing that normality is essential. Our arguments combine resolution of singularities, holonomy techniques, formal completion, and extension properties of holomorphic functions on normal analytic spaces.

A Brown Theorem for Dehn functions of graphs of groups

• Cross-list: math.GR
• Authors: Claudio Llosa Isenrich; Jannis Weis
• arXiv: 2608.07191
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 25 pages

We prove an upper bound on the Dehn function of a group $G$ acting cellularly, cocompactly, and without inversions on a simply connected CW complex $X$ in terms of the Dehn functions of the vertex stabilizers, the Dehn function of $X$, and the distortion of the edge stabilizers, provided that $X$ is either a tree or the stabilizer of each $2$-cell has finite index in the stabilizer of every edge in its boundary. This provides a Dehn function analogue of Brown's Theorem for finiteness properties and an answer to a question of Zaremsky in these cases. We also prove analogues of our result for higher Dehn functions when $X$ is a tree.

Real morsifications via the trace map

• Cross-list: math.AG
• Authors: Pablo Portilla Cuadrado
• arXiv: 2608.07212
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)
• Comments: 35 pages

We prove that every reduced real plane curve singularity admits a real morsification. This settles a question of A'Campo and Gusein-Zade, later stated as conjectures by Leviant--Shustin and by Fomin--Pylyavskyy--Shustin--Thurston. In particular we overcome the main obstruction that remained posed by conjugate pairs of nonreal branches. Our new main ingredient is a construction that produces the divide from a nodal smoothing of two normalization disks. This is what we call the trace map. For real branches, it recovers Gusein-Zade's construction using Chebyshev polynomials. For pairs of complex conjugate branches with distinct tangents, the construction gives an explicit formula for the divide in terms of the Puiseux data. The general method consists in a delicate combination of the trace map with A'Campo's translations and contractions to produce divides and real morsifications for all reduced real plane curve singularities.