math.GT daily digest: 13 submissions for 17 September 2026

New submissions

$\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds

• Authors: Anar Akhmedov
• arXiv: 2609.17603
• Comments: 36 pages, 4 figures
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Symplectic Geometry (math.SG)

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y\_K$ and $X\_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of from , keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework.

For the trefoil block $Y\_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to . A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of , rather than the involutive gluing defining the original $X\_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and . An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to . The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

A curve-based survey of knot Floer homology and concordance invariants

• Authors: Jonathan Hanselman
• arXiv: 2609.17739
• Comments: 58 pages, 30 figures
• Subjects: Geometric Topology (math.GT)

Knot Floer homology associates to a knot in a bigraded chain complex over \mathbb{F}\[W,Z\]!, from which many classical and concordance invariants can be extracted. Recent work shows that this algebraic object can equivalently be represented by a decorated immersed multicurve in a marked surface. This survey explains the immersed curve interpretation of knot Floer homology, aided by many examples, and shows how several invariants arising from the knot Floer complex can be extracted from the corresponding immersed curves. The decorated multicurve associated to a knot has a distinguished curve component $\gamma \_0$ and a distinguished connected component $\Gamma \_0$, both of which are concordance invariants of the knot. We pay particular attention to these components and various numerical concordance invariants that can be extracted from them. We introduce new generalizations of the $V\_s$ invariants, and we give a new curve-based description of the Upsilon invariant by showing that it is determined by generalized $V\_s$ invariants.

Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds

• Authors: Anar Akhmedov; Sümeyra Sakallı
• arXiv: 2609.17871
• Comments: 12 pages, 8 figures
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory

• Authors: Sungkyung Kang; JungHwan Park; Masaki Taniguchi
• arXiv: 2609.18043
• Comments: 49 pages
• Subjects: Geometric Topology (math.GT)

We study -cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi's work, none were previously known to distinguish smooth and topological -cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg--Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth -cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to .

Generating the twist subgroup of the level $2$ mapping class group of a non-orientable closed surface

• Authors: Ryoma Kobayashi
• arXiv: 2609.18433
• Comments: 12 pages, 10 figures
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We give two small generating sets for the subgroup of the mapping class group of a non-orientable closed surface which is generated by Dehn twists and acts trivially on the $\mathbb{Z}\_2$ coefficient first homology group of the surface.

Branched real projective structures on surfaces and geometrisation of representations

• Authors: Gianluca Faraco; Nicholas Rungi
• arXiv: 2609.18436
• Comments: 50 pages, 20 figures and 2 tables
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We introduce and study branched real projective structures on compact surfaces. Our main result shows that every representation of the fundamental group of a compact surface into arises as the holonomy representation of a branched real projective structure, regardless of whether the surface is orientable. We also consider the realisation problem with prescribed branching data, relating the branching degree to the second Stiefel--Whitney class of the representation. The results obtained in this direction suggest several interesting avenues for future research.

Families of knots that cannot be made Legendrian parametrically

• Authors: Javier Martínez-Aguinaga
• arXiv: 2609.18492
• Comments: The results of this article appeared in preliminary form in version 1 of arXiv:2406.04293. The project was later split and these results no longer appeared in version 2 of arXiv:2406.04293. This article presents an improved and expanded version of those results
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Symplectic Geometry (math.SG)

The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the case. We prove that for every , every knot type $\mathcal{K}$, every Legendrian representative $\mathcal{L}$ and every formal Legendrian representative $\mathcal{F}\mathcal{L}$, the associated group homomorphisms and are never surjective. We then show that surjectivity at the $\pi \_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $\pi \_1$.

The flip symmetry on Khovanov-Rozansky homology

• Authors: Hongjian Yang
• arXiv: 2609.18780
• Comments: 38 pages
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky \frak{gl}}\_{N! homology. We prove that this involution is diagonalizable with eigenvalues $\pm 1$. On the one hand, it is the identity over $\mathbb{F}\_2$, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over $\mathbb{Z}$ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar \frak{gl}}\_{N! webs, and (2) the computation of the flip map for planar \frak{gl}}\_{N! webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type $A$ Soergel bimodules, which might be of independent interest.

Cross submissions

Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy

• Authors: Anar Akhmedov
• arXiv: 2609.17634
• Cross-list: math.AG
• Comments: 25 pages, 8 figures
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We compute the geometric monodromy factorization of Xiao's genus-two Lefschetz fibration directly from its branched-cover construction, following Moishezon's braid-monodromy method. The complete quadrangle determines the motion of the six branch points and gives four nonseparating and three separating vanishing cycles. At each of the three $3+3$ degenerations, the branch motion determines the spherical mapping class, and Xiao's local holomorphic model determines its genus-two lift. After fixing a distinguished system of paths, we obtain an ordered positive factorization of type $(4,3)$ and give Artin coordinates for the seven factors.

Integer Realization of an Equivelar Octahedron of Genus 3

• Authors: Ruslan Mizhaev
• arXiv: 2609.17700
• Cross-list: math.CO
• Comments: 5 pages, 2 figures, includes exact integer coordinate tables
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

We present an integer-coordinate realization of a genus-3 polyhedral surface with eight planar simple nonagonal faces. It has 24 vertices and 36 edges, and each vertex is incident with three faces. Every pair of faces shares an edge: 20 pairs share one edge and 8 pairs share two. Vertex coordinates, face walks, and plane equations are given. Exact verification confirms that all faces are planar, the surface is closed and orientable, no unintended intersections occur, and the realization has $C\_4$ symmetry. The construction is based on an earlier construction and preserves its incidence structure.

Entropy and Translation Length in the Ray Graph

• Authors: Juliette Bavard; Danny Calegari; Alden Walker
• arXiv: 2609.17879
• Cross-list: math.DS
• Comments: 13 pages, 4 figures
• Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

Let $\Gamma$ be the mapping class group of the plane minus a Cantor set, acting on the ray graph $\mathcal{R}$, and for let $\tau (\gamma )$ denote the translation length of $\gamma$ on $\mathcal{R}$. We prove that for every diffeomorphism $f$ of representing $\gamma$, where $h$ denotes topological entropy.

The proof falls into two parts: a combinatorial argument bounding distance between two rays in $\mathcal{R}$ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the hypothesis on $f$ can be removed.

The second variation of 2-spheres in the 2n-sphere

• Authors: Gavin Ball; Jesse Madnick
• arXiv: 2609.17887
• Cross-list: math.DG
• Comments: 37 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.

Skein theory, line defects, and quantum symmetric pairs

• Authors: Eric Yen-Yo Chen; David Jordan; Iordanis Romaidis
• arXiv: 2609.18902
• Cross-list: math.QA
• Comments: 32 pages. Comments welcome!
• Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)

We construct skein theory for 3-manifolds with embedded line defects, starting from the data of a ribbon tensor category and its balanced braided module category. We focus on line defects arising from quantum symmetric pairs, and prove finiteness properties for their defect skein modules. As an application, we consider $\mathbf{Z}\_2$-equivariant skein theory and establish an equivalence with defect skein theory in certain settings, leading to a skein theoretical construction of a family of DAHA-modules in the Type AIII case.