math.GT daily digest: 5 new submissions for 24 August 2026

New submissions

Some knots with no SU(2)-abelian surgeries

• Authors: Giacomo Bascape
• arXiv: 2608.20551
• Subjects: Geometric Topology (math.GT)

A knot in is said to be \emph{not} $SU(2)$-abelian knot, if every non-trivial surgery along it yields a $3$-manifold whose fundamental group admits an irreducible $SU(2)$-representation. We provide examples of knots in that are not $SU(2)$-abelian. A knot $K$ is said to be $SU(2)$-\emph{clean} if whenever the fundamental group of the integer $r$-surgery has no $SU(2)$-irreducible representation, then the Alexander polynomial of $K$ does not vanish at any $r$-th root of unity. We show that if $K$ is a non-trivial $SU(2)$-clean knot, the connected sum K\\#K! is never $SU(2)$-abelian. Finally, combining known results on $SU(2)$-abundant knots and a classical epimorphism between knot groups, we show that \emph{every} non-torus knot with at most $9$ crossings is not $SU(2)$-abelian, with at most the two exceptions of $9\_47$ and $9\_49$.

The r-coverings and local moves of planar virtual knotoids

• Authors: Jiacheng An, Fengling Li, Andrei Vesnin
• arXiv: 2608.20624
• Subjects: Geometric Topology (math.GT)
• Comments: 30 pages, 27 figures, comments are welcome!

In this paper, we study planar virtual knotoids, which generalize virtual knots and knotoids both. We consider the problem of whether two given planar virtual knotoids can be transformed into each other via a sequence of local moves and what is the shortest sequence length. By introducing $r$-covering of planar virtual knotoids, we determine the homotopy relationship between different planar virtual knotoids, and obtain lower bounds of the Gordian distance between them. On the basis of these results, we demonstrate calculations of the exact Gordian distance of several given pairs of homotopic planar virtual knotoids. Furthermore, we investigate $Delta$-moves and virtual region crossing changes for planar virtual knotoids, and derive lower bounds of the Gordian distance with respect to both moves via $r$-coverings.

A reconstruction lemma for enhanced bypass sequences

• Authors: Matthias Scharitzer
• arXiv: 2608.20912
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 14 pages, 7 Figures, Comments are welcome!

In the appendix of [LSV26], the authors produce an invariant for contact structures on \\Sigma \\times \[0,1\]!. In this paper, we prove a result announced there which states that this is in-fact a full invariant. As an application, we provide an independent proof of Eliashbergs theorem and show that various construction methods for contact structures are equivalent.

Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds

• Authors: Matthew Cellot
• arXiv: 2608.20959
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Quantum Algebra (math.QA)
• Comments: 46 pages

We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $sigma$ via $sigma$-colorings of triangulations, and we generalize Pachner's theorem in this context.

Cross submissions

Busemann G-spaces with convex balls

• Authors: Tadashi Fujioka, Shijie Gu
• arXiv: 2608.20800
• Cross-list: math.MG
• Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG); Geometric Topology (math.GT)

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii--Halverson--Repovš.