math.GT daily digest: 9 new submissions for 2 September 2026

New submissions

Decompositions and diagrams of symplectic surfaces in Weinstein domains

• Authors: Román Aranda; Patricia Cahn; Agniva Roy; Melissa Zhang
• arXiv: 2609.00163
• Subjects: Geometric Topology (math.GT)
• Comments: 31 pages, 21 figures, comments welcome!

We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi--Piergallini, recovers Islambouli--Starkston's result that every compact Weinstein domain admits a bisection with divides.

A survey on mapping class groups of 3-manifolds

• Authors: Philipp Bader; Rachael Boyd; Giulia Carfora; Gabriel Corrigan; Daniel Galvin; Csaba Nagy; John Nicholson; Weizhe Niu; Isacco Nonino; Mark Pencovitch; Mark Powell
• arXiv: 2609.00233
• Subjects: Geometric Topology (math.GT)
• Comments: 82 pages, 2 figures. Comments welcome

We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.

Multicrossing complex of knot and secant classes

• Authors: Igor Nikonov
• arXiv: 2609.00285
• Subjects: Geometric Topology (math.GT)

The multicrossing homology of a knot can be considered as a generalization of the homology of its fundamental quandle. We show that certain sums of knot secants define invariant classes in the multicrossing homology.

Lipshitz-Sarkar refines mirrors more than Khovanov

• Authors: Yang-Hui He
• arXiv: 2609.00976
• Subjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA)
• Comments: 8 pages

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}\_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of {\it chat-GTP 5.6 sol Pro} combing over the {\it Regina Census} of knots.

Exact curve counting of given word length on the once-punctured torus

• Authors: Filippo Baroni; David Fisac; Mingkun Liu
• arXiv: 2609.01382
• Subjects: Geometric Topology (math.GT); Computational Geometry (cs.CG); Combinatorics (math.CO)
• Comments: 25 pages, 8 figures

On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].

The $\mathrm{FK}\_3$ knot polynomial

• Authors: Stavros Garoufalidis; Shana Yunsheng Li
• arXiv: 2609.01506
• Subjects: Geometric Topology (math.GT)
• Comments: 14 pages

We present the knot polynomial associated to the 12-dimensional sporadic Nichols algebra $\mathrm{FK}\_3$ with automorphism, compute it for all knots with $\le$ 16 crossings and discuss the found patterns. This knot polynomial is not a Vassiliev power series. We ponder how it compares to the knot polynomials that come from Lie algebras and their representations and what the other knot polynomials associated with the higher dimensional sporadic Nichols algebras are.

Homeomorphisms of surfaces in $4$-manifolds

• Authors: Anthony Conway; Daniel Kasprowski
• arXiv: 2609.01545
• Subjects: Geometric Topology (math.GT)
• Comments: 62 pages

This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}\_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group $\mathbb{Z}\_2$ and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in with knot group $\mathbb{Z}\_2$ and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.

Cross submissions

Accessible CAT$(-1)$ groups of critical exponent less than one

• Authors: Yong Hou
• arXiv: 2609.00557
• Cross-list from: Group Theory (math.GR)
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Let $X$ be proper CAT$(-1)$ and let \Gamma\le\Isom(X)! be finitely generated and discrete. The sharp structural theorem states that, if $\Gamma$ is accessible over finite subgroups and , then $\Gamma$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and collapses exactly their conjugate two point boundaries. The result is hereditary, and below $1/2$ every finitely generated subgroup is convex-cobounded (\cref{thm:accessible-main}). Hence non-virtually-free accessible groups have (\cref{cor:accessible-gap}). Consequences cover finitely presented groups, groups with uniformly bounded finite-subgroup orders, characteristic-zero linear groups, and Kleinian groups, also infinite parabolic-free Kleinian groups have finite-index classical Schottky subgroups (\cref{cor:accessibility-extension,cor:linear-groups,cor:kleinian-classical}). Hence we cover substantial larger class than \cite{LiuWang2023},\cite{Hou2001}, also see \cref{rem:strictness-sharpness}. Finally, we also state consequences for finite JSJ representatives and hierarchies (\cref{thm:JSJ,thm:hierarchy,cor:hierarchy-dimension}).

Higher smooth surgery structure sets of complex projective spaces, part II

• Authors: Samuel Kalužný; Tibor Macko
• arXiv: 2609.01505
• Cross-list from: Algebraic Topology (math.AT)
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: 39 pages

We determine the torsion part of the higher smooth surgery structure sets of complex projective spaces (up to some extension problems) and complete the description of the forgetful map to their topological versions in low dimensions. In addition, an application to mapping class groups of complex projective spaces in low dimensions is presented.