math.GT daily digest: 13 submissions for 29 September 2026

New submissions

Lens Space Surgeries and the Bleiler-Litherland Conjecture

• Authors: Qilong Guo
• arXiv: 2609.31757
• Comments: 9 pages, 1 figure
• Subjects: Geometric Topology (math.GT)

We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot $K$ of genus $g$ admits a lens space surgery with slope , then has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.

Purely cosmetic surgeries on knots with low-span Jones polynomials

• Authors: Kazuhiro Ichihara
• arXiv: 2609.32507
• Comments: 16 pages, 1 figure,
• Subjects: Geometric Topology (math.GT)

In this paper, we show that a knot with a nontrivial Jones polynomial of span at most 11 does not admit purely cosmetic surgery.

Satellites and invariants of links

• Authors: Sergey A. Melikhov
• arXiv: 2609.33309
• Comments: 59 pages, 2 figures. To appear in Zap. Nauchn. Sem. POMI (=J. Math. Sci.) About 3 pages of preliminaries migrated from v1-v3 of arXiv:math/0312007 and will be removed from its future versions (starting from v4)
• Subjects: Geometric Topology (math.GT)

An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link is obtained from a link by replacing each knot with its -cable for some and , we have .

The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in which is not a function of the pairwise linking numbers? We solve it affirmatively. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function expanded as a formal power series in Conway's variables .

We also discuss type $n$ invariants which are "cableable up to an invariant of type ", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's -invariants), and applications to links of solenoids.

On the Kashaev--Luo--Vartanov Partition Function for Cusped 3-Manifolds: Asymptotics and a squared norm property

• Authors: Ka Ho Wong
• arXiv: 2609.33922
• Comments: 38 pages, 3 figures
• Subjects: Geometric Topology (math.GT)

We study the asymptotic and TQFT-type properties of the Kashaev--Luo--Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a hyperbolic cone structure, we express the exponential decay rate and the 1-loop term of the partition function in terms of, respectively, the volume and the adjoint twisted Reidemeister torsion of the corresponding hyperbolic cone structure. Consequently, these asymptotic formulas hold for any angle structure having the same peripheral angular holonomies as the geometric one.

We further introduce a Reshetikhin--Turaev-type function associated with certain admissible Neumann--Zagier data, extending the Jones function in Teichmüller TQFT previously studied by Ben Aribi and the author. We show that, after a combinatorial normalization, for any triangulation, the KLV partition function can be expressed as a weighted integral of the squared norm of the Reshetikhin--Turaev-type function, where the weight is determined by the angular holonomies of peripheral curves. In particular, for FAMED triangulations, the Reshetikhin--Turaev-type function agrees with the Jones function, so that the normalized KLV partition function is obtained by integrating the squared norm of the integrand defining the Teichmüller TQFT partition function. This provides a noncompact analogue of the relationship between Turaev--Viro and Reshetikhin--Turaev invariants.

Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

• Authors: Soichiro Uemura
• arXiv: 2609.34108
• Comments: 41 pages, 4 figures
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.

Exceptional pretzel knots are not algebraically slice

• Authors: Suman Saurabh
• arXiv: 2609.34522
• Comments: 8 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

We prove that the Alexander polynomial of every knot in Lecuona's exceptional family of pretzel knots fails the Fox--Milnor condition. Consequently, none of these knots is algebraically or topologically slice. Together with work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots. It also removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots up to reordering of parameters.

On pure braids and bonded braids

• Authors: Sofia Lambropoulou; Lucrezia Beatrice Lorenzi
• arXiv: 2609.35340
• Comments: 45 pages, 31 figures
• Subjects: Geometric Topology (math.GT)

In this paper we investigate the relation between the bonded braid monoid BBn and the group of pure braids , realised by an algebraic analogue of the topological tangle insertion, namely by replacing a bond by (a power of) a pure braid generator. To do so, we provide appropriate presentations for using cyclic relations and clasp generators, and extend them to presentations of . We construct the monoid epimorphism , mapping bonds to pure braid generators, which is the main result of the paper. Using the isomorphism from the singular braid monoid to the tight bonded braid monoid , we extend $\Phi$ to a new type of epimorphism . The epimorphisms $\Phi$ and $s\Phi$ are extended to families of epimorphisms , , .

Cross submissions

Motion planning invariants and families of subgroups

• Authors: Ekansh Jauhari; Ben Knudsen
• arXiv: 2609.30579
• Cross-list: math.AT
• Comments: 18 pages. Comments are welcome
• Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

Equivariant isometric immersions of surfaces in hyperbolic space

• Authors: Jean-Marc Schlenker
• arXiv: 2609.32300
• Cross-list: math.DG
• Comments: 26 pp
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature , and let c\in \cT_S! be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in \HH^3! such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if is a smooth metric on $S$ with curvature and if c\in \cT_S!, there exists a unique equivariant immersion of $S$ into \HH^3! with third fundamental form and such that the pull-back of the conformal class at infinity by the Gauss map is $c$.

Equivalently, given $h$ and c\in \cT_S!, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given and $c$, there exists a unique pair , where $E$ is a hyperbolic end with conformal structure at infinity $c$, and is an embedding inducing the third fundamental form .

Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

Log-concavity of flat arrangement polynomials

• Authors: Yuan Gao; Tianyu Yuan
• arXiv: 2609.33651
• Cross-list: math.CO
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof uses a new mixed-volume representation of the coefficients and the Alexandrov--Fenchel inequality. A more general formula gives a factorization and log-concavity for related mixed-volume sequences. As applications, we establish the conjectured log-concavity for spanning-tree polynomials of Eulerian digraphs, extend it to positive circulation weights, and strengthen the coefficient inequalities for Alexander polynomials of special alternating links.

The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5

• Authors: Zoltán Szabó
• arXiv: 2609.34042
• Cross-list: math.AG
• Comments: 24 pages, 23 figures
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We determine the vanishing cycles of Xiao's genus-two fibrations , , whose fibers admit degree-$d$ maps to a fixed elliptic curve $E$. We tile the base by triangles and realize the surface as a degree-$d$ branched cover of . The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For , these give new positive factorizations of the identity in the genus-two mapping class group, of types $(6,7)$ and $(12,19)$.

Borel complexity for product of trees and commuting partial maps

• Authors: Koichi Oyakawa
• arXiv: 2609.34068
• Cross-list: math.GR
• Comments: 11 pages
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Logic (math.LO)

We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.

Braid Stability and a Floer theory of Braid Isotopies

• Authors: Nicolas Grunder
• arXiv: 2609.34961
• Cross-list: math.SG
• Comments: 67 pages, 15 figures, comments are welcome
• Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)

We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs.

With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every there is a sequence of Hamiltonian diffeomorphisms on the two-torus such that $$d_H(\operatorname{Ent}_{\leq \alpha}(T^2,\omega),\phi_k)\to \infty \quad (k \to \infty),$$ where denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $\alpha$ and the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.