math.GT daily digest: 9 submissions for 14 September 2026

New submissions

Holomorphic realizations of pairs of foliations on Riemann surfaces

• Authors: Nathaniel Sagman; Dragomir Saric
• arXiv: 2609.12217
• Subjects: Geometric Topology (math.GT); Complex Variables (math.CV); Differential Geometry (math.DG)

Let $X$ be a hyperbolic Riemann surface and let $\mu$ and $\nu$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $\mu$ and $\nu$ are filling if and only if there exists a homeomorphism to another Riemann surface and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.

Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case

• Authors: Hyungseok Jung
• arXiv: 2609.12295
• Comments: 4 figures, 27 pages. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov--Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index~$3$. Every amphichiral knot of braid index~$3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word~$c$. Building on the Birman--Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on~$c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index~$3$, this yields a complete criterion: is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.

Quantum Invariants indexed by Fibered Faces of the Thurston Polytope

• Authors: Davide Passaro; Lara San Martín Suárez
• arXiv: 2609.12311
• Comments: 58 pages, 16 figures, 9 tables
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We study the Gukov--Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin--Morton--Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yield a family of multivariable Gukov--Manolescu series, indexed by monomials of the Alexander polynomial. We conjecture that these monomials correspond precisely to the fibered faces of the Thurston norm ball and provide extensive computational evidence for this correspondence. Further results concerning the leading term, the corresponding single-variable invariant, and the effect of partial Dehn surgery are also established.

Non-Hitchin Borel Anosov representations from surface groups to

• Authors: Zhufeng Yao; Junming Zhang
• arXiv: 2609.12809
• Comments: 33 pages, including an appendix and a declaration of AI use, comments are very welcome
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS)

We use the Labourie--Wentworth's formula and the thermodynamic formalism to show that, along the slice constructed by Bronstein--Davalo, the logarithmic top-eigenvalue length spectrum has a uniformly positive second variation near the Barbot representation.

As a major application, we show that every closed surface group admits a non-Hitchin Borel Anosov representation into for every . In particular, we obtain the first such examples in the even dimensions $6k$. We also study the local behavior of related objects of this slice around the Barbot representation, including the Lyapunov exponent of the flat bundle, the Hausdorff dimension of the limit set, and the Hilbert entropy of the representation.

How natural of a geometric operation is Murasugi sum?

• Authors: Thomas Kindred
• arXiv: 2609.13093
• Comments: 32 pages, 34 figures. The first 19 pages of this paper were previously part of arXiv:2408.16948. Comments welcome!
• Subjects: Geometric Topology (math.GT)

Gabai proved that any Murasugi sum of -essential Seifert surfaces is also -essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.)

We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.

Knot Floer homology of boundary Dehn twists

• Authors: Rithwik Susheel Vidyarthi
• arXiv: 2609.13097
• Comments: 12 pages, 7 figures. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We prove that the knot Floer complex, along with some restrictions on the flip map, determines the dual knot to $\pm 1$ surgery on the Borromean knot. In particular, this implies that Heegaard Floer homology detects boundary Dehn twists.

Cross submissions

A Gap Theorem For the Mobius Cross Energy

• Authors: Ronggang Li; Chuanhuan Li
• arXiv: 2609.12102
• Cross-list: math.DG
• Comments: 23 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We prove that the critical value is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere . Specifically, there exists such that any non-split, regular pair of curves with disjoint images, having vanishing first variation and energy at most , must lie in the Möbius-reparametrization orbit of the standard Hopf link.

Property $(T)$ and nonlinearity of mapping class group quotients

• Authors: Piotr W. Nowak
• arXiv: 2609.12196
• Cross-list: math.GR
• Comments: 38 pages
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We give a general method for proving Kazhdan's property $(T)$ for quotients , where $G$ is countable, $K$ is a normal subgroup and denotes its lower central series. The method combines an affine realization of , a contraction argument, and permanence for nilpotent normal subgroups. We apply it to the Torelli lower-central quotients for and show that they have property $(T)$ for every . We also prove property $(T)$ for for every , relating these groups to the tame nilpotent images studied by Lubotzky and Pak. For the Torelli lower-central quotients with and , every finite-dimensional complex representation has infinite kernel, and these quotients are not linear over any field.

The Morita classes are nonzero

• Authors: Alexander Kupers; Jeremy Miller; Peter Patzt
• arXiv: 2609.12951
• Cross-list: math.AT
• Comments: 10 pages
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT); Quantum Algebra (math.QA)

We prove that all Morita classes are nonzero using a novel description of the Hopf algebra structure on Steinberg homology. We prove analogous results with twisted coefficients. Our results disprove a case of a finiteness conjecture of Kontsevich.