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On asymmetric hyperbolic L-space knots of braid index four
Kenneth L. Baker, Masakazu Teragaito
https://arxiv.org/abs/2510.07618 https://arxiv.org/pdf/2510…

On asymmetric hyperbolic L-space knots of braid index four
A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space. In the SnapPy census, there are exactly 9 asymmetric L-space knots. Among them, the knot t12533 is the only known example of braid index 4. We generalize this knot, and give the first infinite family of asymmetric hyperbolic L-space knots of braid index 4.

Trisected Rainbows and Braids
Rom\'an Aranda, Scott Carter, Julia Courtney, Puttipong Pongtanapaisan
https://arxiv.org/abs/2510.04248 https://arxiv.org…

Trisected Rainbows and Braids
New explicit procedures for passing among triplane diagrams, braid movies, and braid charts for knotted surfaces in $\mathbb{R}^4$ are presented. To this end, rainbow diagrams, which lie between braid charts and triplanes, are introduced. Inequalities relating the braid index and the bridge index of 2-knots are obtained via these procedures. Another consequence is a 4-dimensional version of the classical result that ``the minimal number of Seifert circles equals the braid index of a link'' due …

Simple $B_4$ representations associated to cyclotomic Hecke algebras
Lilit Martirosyan, Hans Wenzl
https://arxiv.org/abs/2510.09928 https://arxiv.org/pdf/2…

Simple $B_4$ representations associated to cyclotomic Hecke algebras
We determine the structure of the cyclotomic Hecke algebra corresponding to the complex reflection group $G_{25}$ also when it is not semisimple, as long as the generators are diagonalizable. In particular, we classify all simple representations of the braid group $B_4$ for which the generators are diagonalizable and satisfy a cubic polynomial. This will be used in the classification of braided tensor categories of type $G_2$.

Representation asymptotics in the homology of pure graph braid groups
Louis Hainaut, Ben Knudsen, Nicholas Wawrykow
https://arxiv.org/abs/2510.00201 https://

Representation asymptotics in the homology of pure graph braid groups
We give explicit formulas for the asymptotic Betti numbers, over an arbitrary field, of the ordered configuration spaces of a graph. In characteristic zero, we further give explicit formulas for the asymptotic multiplicities in homology of many irreducible representations of the symmetric group, in the spirit of representation stability.

Natural transformations between braiding functors in the Fukaya category
Yujin Tong
https://arxiv.org/abs/2511.10462 https://arxiv.org/pdf/2511.10462 https://arxiv.org/html/2511.10462
arXiv:2511.10462v1 Announce Type: new
Abstract: We study the space of $A_\infty$-natural transformations between braiding functors acting on the Fukaya category associated to the Coulomb branch $\mathcal{M}(\bullet,1)$ of the $\mathfrak{sl}_2$ quiver gauge theory. We compute all cohomologically distinct $A_\infty$-natural transformations $\mathrm{Nat}(\mathrm{id}, \mathrm{id})$ and $\mathrm{Nat}(\mathrm{id}, \beta_i^-)$, where $\beta_i^-$ denotes the negative braiding functor. Our computation is carried out in a diagrammatic framework compatible with the established embedding of the KLRW category into this Fukaya category. We then compute the Hochschild cohomology of the Fukaya category using an explicit projective resolution of the diagonal bimodule obtained via the Chouhy-Solotar reduction system, and use this to classify all cohomologically distinct natural transformations. These results determine the higher $A_\infty$-data encoded in the braiding functors and their natural transformations, and provide the first step toward a categorical formulation of braid cobordism actions on Fukaya categories.
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BRAID: Input-Driven Nonlinear Dynamical Modeling of Neural-Behavioral Data
Parsa Vahidi, Omid G. Sani, Maryam M. Shanechi
https://arxiv.org/abs/2509.18627 https://

BRAID: Input-Driven Nonlinear Dynamical Modeling of Neural-Behavioral Data
Neural populations exhibit complex recurrent structures that drive behavior, while continuously receiving and integrating external inputs from sensory stimuli, upstream regions, and neurostimulation. However, neural populations are often modeled as autonomous dynamical systems, with little consideration given to the influence of external inputs that shape the population activity and behavioral outcomes. Here, we introduce BRAID, a deep learning framework that models nonlinear neural dynamics un…

Canonical Reduction Systems in Artin-Tits groups of spherical type
Mar\'ia Cumplido, Juan Gonz\'alez-Meneses, Davide Perego
https://arxiv.org/abs/2510.00713 https://

Canonical Reduction Systems in Artin-Tits groups of spherical type
We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we p…

How large is the braid monodromy of low-genus Lefschetz fibrations?
Faye Jackson
https://arxiv.org/abs/2510.04389 https://arxiv.org/pdf/2510.04389

How large is the braid monodromy of low-genus Lefschetz fibrations?
Given a genus $g$ smooth Lefschetz fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, we describe the subgroup $\operatorname{Br}(π)$ of the spherical braid group $\operatorname{Mod}(S^2,Δ)$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\operatorname{Mod}(S^2,Δ) : \operatorname{Br}(π)]$ is infinite. As an application of our methods, we prove that $[\operatorname{Mod}(S^2,Δ) : \operato…

Quantum queer superalgebra and its integral form
Jianmin Chen, Zhenhua Li, Hongying Zhu
https://arxiv.org/abs/2509.24265 https://arxiv.org/pdf/2509.24265…

Quantum queer superalgebra and its integral form
In this paper, we introduce quantum root vectors for the quantum queer superalgebra ${\boldsymbol U}_{\!{v}}({\mathfrak q_n})$ via a braid-group action, compute their complete commutation relations, and construct a PBW-type basis for the Lusztig integral form ${{\boldsymbol U}_{v,\mathcal{ Z}}}$. This yields an explicit presentation of ${{\boldsymbol U}_{v,\mathcal{ Z}}}$ and provides a way to understand the structure of the quantum queer superalgebra at roots of unity.

Continuous approximate roots of polynomial equations via shape theory
Joshua Lau, Vicente Marin-Marquez
https://arxiv.org/abs/2510.01613 https://arxiv.org/…

Continuous approximate roots of polynomial equations via shape theory
We study continuous approximate solutions to polynomial equations over the ring $C(X)$ of continuous complex-valued functions over a compact Hausdorff space $X$. We show that when $X$ is one-dimensional, the existence of such approximate solutions is governed by the behaviour of maps from the fundamental pro-group of $X$ into braid groups.

Extremal monomials of $q$-characters
Andrei Negu\c{t}
https://arxiv.org/abs/2510.08103 https://arxiv.org/pdf/2510.08103

Extremal monomials of $q$-characters
In this short paper, we prove a conjecture of Frenkel-Hernandez, which states that $q$-characters of finite-dimensional simple modules of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ are bounded by the Weyl group orbit of the leading monomial under Chari's braid group action. This generalizes the Weyl group invariance of characters of finite-dimensional representations of $\mathfrak{g}$.

The 3-strand braid group with torsion
Ethan Dlugie, Tahsin Saffat
https://arxiv.org/abs/2509.17900 https://arxiv.org/pdf/2509.17900

The 3-strand braid group with torsion
In the 1950s, H. S. M. Coxeter considered the quotients of braid groups given by adding the relation that all half Dehn twist generators have some fixed, finite order. He found a remarkable formula for the order of these groups in terms of some related Platonic solids. Despite the inspiring apparent connection between these "truncated" braid groups and Platonic solids, Coxeter's proof boils down to a finite case check that reveals nothing about the structure present. We give a topological inter…

Monodromy of supersolvable toric arrangements
Christin Bibby, Daniel C. Cohen, Emanuele Delucchi
https://arxiv.org/abs/2510.00166 https://arxiv.org/pdf/251…

Monodromy of supersolvable toric arrangements
We study topological aspects of supersolvable abelian arrangements, toric arrangements in particular. The complement of such an arrangement sits atop a tower of fiber bundles, and we investigate the relationship between these bundles and bundles involving classical configuration spaces. In the toric case, we show that the monodromy of a supersolvable arrangement bundle factors through the Artin braid group, and that of a strictly supersolvable arrangement bundle factors further through the Arti…

Replaced article(s) found for math.GR. https://arxiv.org/list/math.GR/new
[1/1]:
- Membership problems in braid groups and Artin groups
Robert D. Gray, Carl-Fredrik Nyberg-Brodda

Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$
Joe Boninger
https://arxiv.org/abs/2509.15321 https://arxiv.org/pdf/2…

Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$
The Burau representation of braid groups and knot Floer homology share a link to the Fox calculus. We make this connection explicit, with the following outcome: if $B$ is the full Burau matrix of any braid, and $A$ is any square submatrix of $B - λI$, we define a Heegaard Floer homology theory that categorifies $\det(A)$ and is an invariant of the braid. We also describe an analogous construction for the Gassner representation. Then, we leverage the relationship between the Burau representatio…

Quantum Howe duality and Schur duality of type AIII
Weinan Zhang
https://arxiv.org/abs/2509.18982 https://arxiv.org/pdf/2509.18982

Replaced article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- How large is the braid monodromy of low-genus Lefschetz fibrations?
Faye Jackson

Ping-pong for basis-conjugating HNN-extension of free group
Vasily Ionin
https://arxiv.org/abs/2509.19635 https://arxiv.org/pdf/2509.19635

Ping-pong for basis-conjugating HNN-extension of free group
We construct a normal form for a multiple HNN-extension of a free group by basis-conjugating embeddings. We provide sufficient conditions on a collection of subgroups to fulfill the requirements of ping-pong lemma. Recall that the pure braid group splits as a semidirect product of free groups. Using our result, we show that certain braids from the first two summands $F_n \rtimes F_{n-1} \subset P_{n+1}$ generate a free subgroup.

Replaced article(s) found for math.QA. https://arxiv.org/list/math.QA/new
[1/1]:
- A Categorical Perspective on Braid Representations
P. P. Martin, E. C. Rowell, F. Torzewska

Crosslisted article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- Representation asymptotics in the homology of pure graph braid groups
Louis Hainaut, Ben Knudsen, Nicholas Wawrykow

On the arc index and Turaev genus of a link
\'Alvaro Del Valle V\'ilchez, Adam M. Lowrance
https://arxiv.org/abs/2509.21274 https://arxiv.org/pdf/2…

On the arc index and Turaev genus of a link
We compute the arc index of an adequate link and establish bounds on the arc index of the closure of a positive 3-braid. We also conjecture an inequality between the crossing number, arc index, and Turaev genus of a link and show the conjecture is true for several infinite families of links including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots.

Lefschetz pencils on a complex projective plane from a topological viewpoint
Ju A Lee
https://arxiv.org/abs/2509.21069 https://arxiv.org/pdf/2509.21069

Lefschetz pencils on a complex projective plane from a topological viewpoint
In this article, we present a differential topological construction of symplectic Lefschetz pencils of genus $\frac{(d-1)(d-2)}{2}$ with $d^2$ base points and $3(d-1)^2$ critical points for arbitrary $d\geq 4$, analogous to the holomorphic Lefschetz pencils of curves of degree $d$ in $\mathbb{C}P^2$. Moreover, for the case $d=4$, we derive an explicit monodromy factorization of the genus $3$ holomorphic Lefschetz pencil on $\mathbb{C}P^2$ based on the braid monodromy technique and prove that it…