arXiv · 2610.06429
We give a Seifert-form criterion for upper bounds on topological shaking numbers and prove that isotropic certificates lift under $(m,1)$-cabling.
Explore connections, maps & timelines
Weizhe Shen. 2026-10-05. Topological Shaking Numbers and Cabling. https://arxiv.org/abs/2610.06429
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
A nilpotent quandle is a quandle whose inner automorphism group is nilpotent. Such quandles have been called reductive in previous works, but it turns out that their behaviour is in fact very close to nilpotency for groups. In particular, we show that it is easy to characterise generating sets of such quandles, and that they have the Hopf property. We also show how to construct free nilpotent quandles from free nilpotent groups. We then use the properties of nilpotent quandles to describe a simple presentation of their associated group, and we use this to recover the classification of abelian quandles by Lebed and Mortier [LM21]. We also study reduced quandles, and we show that the reduced fundamental quandle is equivalent, as an invariant of links, to the reduced peripheral system, sharpening a previous result of Hughes [Hug11]. Finally, we give a characterisation of nilpotency in terms of the associated invariants of braids.
In this paper, we examine the behavior of twisted Reidemeister torsion for compact, connected, oriented $3$-manifolds under the connected sum operation. Under appropriate assumptions, we establish a multiplicative relation for the torsion associated with representations into a complex reductive algebraic group. Our results are consistent with the known behavior of torsion-type invariants and provide a formulation that may be useful for further investigations of decomposition properties in $3$-manifold topology.
In his shire theorem, G. Pólya proves that the zeros of iterated derivatives of a meromorphic function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. By recasting the local arguments of Pólya into the language of translation surfaces, we prove its generalisation describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator $T_ω: f \mapsto \frac{df}ω$ where $ω$ is a given meromorphic $1$-form. The accumulation set of these zeros is the union of edges of a generalised Voronoi diagram defined by the initial function $f$ together with the singular flat metric on the Riemann surface induced by $ω$. This result provides the ground for a novel approach to the problem of finding a flat geometric presentation of a translation surface initially defined in terms of algebraic or complex-analytic data.