Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.06752

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Hamid Abchir·Khaled Qazaqzeh·Mohammed Sabak
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study alternateness of the Jones-Kauffman polynomial of virtual links. In particular, we identify a class of virtual links with alternating Jones-Kauffman polynomial that will be called the class of mock-alternating virtual links. Then, we explain how every mock-alternating virtual link can give rise to an infinite family of mock-alternating virtual links using the twisting technique introduced in [4] and later generalized in [5,18]. Finally, we give a connected state expansion of the mock-determinant for checkerboard-colorable links.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hamid Abchir, Khaled Qazaqzeh, Mohammed Sabak. 2026-10-05. On Alternateness of the Jones-Kauffman Polynomial of Virtual Links. https://arxiv.org/abs/2610.06752

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Nilpotent quandles

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.

math.GT↗
arXiv2026

On twisted torsion of compact $3$-manifolds

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.

math.GT↗
arXiv2026

The translation geometry of Pólya's shires

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.

math.GT↗