math.GT daily digest: 8 new submissions for 5 August 2026

New submissions

Splitting spheres for -links in

• Authors: Jianfeng Lin; Yi Xie; Boyu Zhang
• arXiv: 2608.02785
• Subjects: Geometric Topology (math.GT)
• Comments: 13 pages, 2 figures. Comments are welcome!

We prove that every smooth two-component split sphere link admits infinitely many smooth splitting $3$-spheres that are topologically non-isotopic. This generalizes a theorem of Tatsuoka from the two-component sphere unlink to split links with arbitrarily knotted sphere components. In the course of the proof, we establish a general sufficient condition under which a connected sum of smooth $4$-manifolds admits infinitely many topologically non-isotopic splitting $3$-spheres. This criterion may be of independent interest; in particular, it applies to all previously known examples of nonuniqueness for splitting $3$-spheres of positive-genus surface links.

Barbell twists are natural

• Authors: Yi Liu
• arXiv: 2608.02801
• Subjects: Geometric Topology (math.GT)
• Comments: 32 pages; comments welcome

For any oriented smooth $4$--manifold $X$ diffeomorphic to (), the author establishes a natural isomorphism of abelian groups:

[ \mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}), ]

concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney--Gabai barbell twist is identified with a generator of the factor subgroup . Up to boundary-fixing diffeotopy, the barbell spines of $\mathcal{N}$ are completely classified by the bases of , forming a homogeneous set modeled on the group . Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $\phi$ or in , according to the sign of the homological basis orientation.

Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots

• Authors: Michal Jablonowski
• arXiv: 2608.03818
• Subjects: Geometric Topology (math.GT)
• Comments: 21 pages

For every admissible pair of Brieskorn parameters in the Plotnick--Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every , a family of $N$ oriented $2$-knots with these properties.

We additionally show that the Tanaka--Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.

Unbounded Gaps Between Ordinary and Equivariant Dehn Surgery Numbers

• Authors: Qilong Guo; Chunxing Yan
• arXiv: 2608.03886
• Subjects: Geometric Topology (math.GT)
• Comments: 12 pages, no figures

For a closed oriented $3$-manifold $Y$ and an orientation-preserving involution $\tau$, let \DS(Y)! denote the minimum number of components in an integral surgery description of $Y$, and let \EDS(Y,\tau)! denote the corresponding minimum among periodic surgery descriptions inducing $\tau$. We prove that for every integer there is a pair such that

[ \DS(Y_k)=k, \qquad \EDS(Y_k,\tau_k)=2k. ]

Consequently, the difference \EDS(Y,\tau)-\DS(Y)! is unbounded even when $\tau$ is an involution. This answers Problems~1.15(b) and~1.15(c) in the K3 problem list. We also construct infinitely many pairwise nonhomeomorphic irreducible lens spaces $Z$ admitting involutions $\sigma$ for which

[ \DS(Z)<\EDS(Z,\sigma). ]

The next-to-top term of the knot Floer homology of some non-fibered knots

• Authors: Fraser Binns; Shunyu Wan
• arXiv: 2608.03900
• Subjects: Geometric Topology (math.GT)
• Comments: 29 pages, 8 figures, comments welcome

Sivek conjectured that the rank of knot Floer homology in the next-to-top Alexander grading is at least the rank in the top Alexander grading. Baldwin and Vela-Vick verified this conjecture in the case of fibered knots arXiv:1801.06563. Ni gave a generalization of this result (for knots in generalized $L$-spaces) to cases in which the knot Floer homology satisfies an algebraic condition arXiv:2104.14687. We give an independent generalization of Baldwin and Vela-Vick's result to a family of knots with Seifert surfaces satisfying certain conditions.

Cross submissions

Zesting and the relative complexity of Reshetikhin-Turaev invariants

• Cross-list: math.QA
• Authors: Colleen Delaney; Calvin McPhail-Snyder
• arXiv: 2608.02795
• Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Geometric Topology (math.GT)
• Comments: 44 pages, many figures

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $\zeta$ to produce a new category whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links depending only on $\zeta$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute \emph{tangle} invariants , which leads to a polynomial time algorithm to compute invariants of links . A byproduct of our construction is an identification (up to a sign) of the link invariants as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

Rank 2 Affine Invariant Subvarieties in H(6)

• Cross-list: math.DS
• Authors: Pramana Saldin; Ruocheng Yang
• arXiv: 2608.02855
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 25 pages, 15 figures

We classify rank 2 rel 0 arithmetic affine invariant subvarieties in the minimal stratum H(6). The proof follows and extends Apisa's approach in genus three, reducing the analysis to the classification of rank 1 rel 1 cylinder rigid affine invariant subvarieties in lower-genus boundary strata. In particular, together with the classification of algebraically primitive rank 2 rel 0 orbit closures, this gives a complete description of rank 2 affine invariant subvarieties in H(6).

A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra

• Cross-list: math.PR
• Authors: Huabin Ge; Yangxiang Lu; Chuwen Wang; Tian Zhou
• arXiv: 2608.03575
• Subjects: Probability (math.PR); Differential Geometry (math.DG); Geometric Topology (math.GT)

We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in . By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number , determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula . Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when , and hyperbolic when . Thus, global conformal type is governed by the expectation of a local geometric quantity.

We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits.

To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.