math.GT daily digest: 7 new submissions for 7 August 2026

New submissions

Building Foliations from Heegaard Diagrams

• Authors: Shangjun Shi; Yanqing Zou
• arXiv: 2608.05542
• Subjects: Geometric Topology (math.GT)
• Comments: 15 page, 16 figures

Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation.

Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks.

We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.

Lorenz Links that are not Horseshoe and Rossler Links

• Authors: Thiago de Paiva; Yi Liu
• arXiv: 2608.05562
• Subjects: Geometric Topology (math.GT)
• Comments: 26 pages, 2 figures

We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, Rössler links and horseshoe links are treated as belonging to the same template class.

We show that the relationship between Lorenz links and horseshoe/Rössler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot , with (p, with (p<q), is a horseshoe knot, then (3p\leq 2q). We show that this phenomenon is much broader. We extend the Holmes--Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not Rössler links.

Characterizing slopes for Legendrian knots

• Authors: Youlin Li; Chi Zhang
• arXiv: 2608.06079
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 16 pages, 5 figures

We establish a criterion relating smooth and contact characterizing slopes under a uniqueness assumption. Let $L$ be a Legendrian representative of a knot with standard contact structure, and assume that the isotopy class of $L$ is uniquely determined by its classical invariants: the Thurston--Bennequin invariant $tb(L)$ and the rotation number. Then, for any non-zero rational number $r$, if is a smooth characterizing slope for $K$, it becomes a contact characterizing slope for $L$. As applications, we study the characterizing slopes for Legendrian representatives of the unknot, trefoil, figure-eight knot, cinquefoil, , and .

Fractional Dehn twist coefficients and rank bounds for categorified link invariants

• Authors: Fraser Binns. Diana Hubbard
• arXiv: 2608.06201
• Subjects: Geometric Topology (math.GT)
• Comments: 32 pages, 15 figures, comments welcome!

We give two new lower bounds on the rank of categorified link invariants: one on the link Floer homology of fibered links in terms of the fractional Dehn twist coefficients of their monodromies, and another, as a corollary, on the annular Khovanov homology of braid closures in terms of the fractional Dehn twist coefficient of the braid. The most important technical component of the proof is that we determine the behaviour of the link Floer homology of fibered links under adding boundary Dehn twists to their monodromies in the next to top Alexander grading.

Cross submissions

Topological Characterizations of Geometrically Infinite Convergence Group Actions and Orbit Uniform Metrics

• Cross-list: math.GR
• Authors: Chaodong Yang
• arXiv: 2608.05311
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 18 pages, 1 figure

Two characterizations of geometric finiteness in terms of weaker conditions are known for actions on Gromov hyperbolic spaces. In this paper, we establish analogous characterizations for general convergence group actions. To this end, we introduce the notion of an orbit uniform metric. We prove that a point is either conical or bounded parabolic if and only if its orbit is discrete with respect to an orbit uniform metric. As a consequence, we characterize geometric infiniteness in terms of the uncountability of the set of non-conical limit points. We further characterize geometric infiniteness by the existence of escaping sequences of hyperbolic elements, thereby extending the corresponding result for actions on Gromov hyperbolic spaces.

Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants

• Cross-list: math.CO
• Authors: Colin Defant; Nathan Williams
• arXiv: 2608.06225
• Subjects: Combinatorics (math.CO); Group Theory (math.GR); Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 65 pages, 6 figures, 2 tables

Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. We introduce a technique that relates the dual braid group generators, the Hecke images of pure braids, and factorization problems in reflection groups to knot invariants. We prove that the -HOMFLYPT polynomial can be computed as a solution to such a problem. This technique was motivated by Coxeter--Catalan combinatorics. For example, we give a new proof of EL-shellability of the noncrossing partition lattice using the image of the full twist in the Hecke algebra; in a surprising sort of combinatorial reciprocity, its inverse computes the homotopy type. Similarly, noncrossing partitions naturally arise from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword models for rational noncrossing Catalan objects. Our reciprocity gives two models: one generalizing noncrossing partitions, and one generalizing clusters. Applying the same method produces two (rational) noncrossing parking models.

Squarefree Matrix Formulas for the CWR Invariant of Alternating Knots and Links

• Cross-list: math.CO
• Authors: Michal Jablonowski
• arXiv: 2608.06372
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 24 pages

We give weighted-matrix formulas for the components of the $CWR$ invariant of oriented non-split alternating links. After recalling the known trace formulas for and , we give a construction uniform in $k$: attaching an independent commuting variable to each vertex of a consolidated Tait graph and extracting the squarefree part of the resulting trace isolates simple cycles from closed walks. This yields a formula for for every , a log-determinant generating polynomial for each of the two Tait graphs, and an equivalent Moebius-inversion formula over principal submatrices.

Specializing the uniform formula, we obtain explicit closed weighted formulas for and . We also record a bipartiteness criterion for the vanishing of all odd components and a characteristic-polynomial formula for the unweighted specialization of the first nonvanishing odd component. The graph-theoretic constructions apply to arbitrary finite simple loopless weighted graphs; the alternating-link hypothesis enters through the invariance theorem for $CWR$.