Tootfinder

Liftable braids and the coloured braid groupoid
Joan Licata, Vera V\'ertesi
https://arxiv.org/abs/2508.05146 https://arxiv.org/pdf/2508.05146

Liftable braids and the coloured braid groupoid
When $π:\widetildeΣ\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $β$ \textit{lifts} if there is a homeomorphism $\widetildeβ\in \textit{Mod}(\widetildeΣ)$ such that $β\circ π=π\circ \widetildeβ$. For arbitrary covers, the \textit{lifting homomorphism} taking $β$ to $\widetildeβ$ is only defined on a proper subgroup of the braid group. This paper extends the li…

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 …

The discrete periodic Pitman transform: invariances, braid relations, and Burke properties
Eva R. Engel, Benjamin Jasper Kra-Caskey, Oleksandr Lazorenko, Caio Hermano Maia de Oliveira, Evan Sorensen, Ivan Wong, Ryan Xu, Xinyi Zhang
https://arxiv.org/abs/2508.05603

The discrete periodic Pitman transform: invariances, braid relations, and Burke properties
We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. As a corollary, we prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform, shown by Biane, Bougerol, and O'C…

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…

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.

Conjugacy invariants for powers of rigid braids: a uniformity phenomenon
Matthieu Calvez (IRISIB), Juan Gonz\'alez-Meneses (DALG-US), Bert Wiest (IRMAR)
https://arxiv.org/abs/2509.01361

Conjugacy invariants for powers of rigid braids: a uniformity phenomenon
Consider an $m$-strand braid $x$ which is rigid in the sense of Garside-theory. Let $SC(x)$ be the set of rigid conjugates of $x$ -- this is a well-known characteristic subset of the conjugacy class of $x$. We present computational evidence that the sequence $(|SC(x^n)|)\_{n\in\mathbb N}$ is not only bounded, but in fact periodic, and that the length of the period can be bounded in terms of the number of strands $m$. We prove this result in the special case of the 3-strand braid group…

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

Decompositions of augmentation varieties via weaves and rulings
Johan Asplund, Orsola Capovilla-Searle, James Hughes, Caitlin Leverson, Wenyuan Li, Angela Wu
https://arxiv.org/abs/2508.20226

Decompositions of augmentation varieties via weaves and rulings
The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. We prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse compl…

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…

Relative braid group symmetries on modified $\mathrm{i}$quantum groups and their modules
Weiqiang Wang, Weinan Zhang
https://arxiv.org/abs/2508.12041 https://

Relative braid group symmetries on modified $\mathrm{i}$quantum groups and their modules
We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum gr…

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…

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.

The linear minimal 4-chart with three crossings
Teruo Nagase, Akiko Shima
https://arxiv.org/abs/2509.04114 https://arxiv.org/pdf/2509.04114

The linear minimal 4-chart with three crossings
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $Γ$ be a chart, and we denote by $Cross(Γ)$ the set of all the crossings of $Γ$, and we denote by $Γ_m$ the union of all the edges of label $m$. For a 4-chart $Γ$, if the closure of each connected component of t…

On the irreducibility of singular braid group representations
Mohamad N. Nasser
https://arxiv.org/abs/2508.11580 https://arxiv.org/pdf/2508.11580

On the irreducibility of singular braid group representations
We study the irreducibility of complex representations of the singular braid group of $n$ strands, namely $SB_n$. Our first finding is that we determine the forms of all irreducible representations $ρ: SB_2 \rightarrow GL_2(\mathbb{C})$. The second finding is that we determine the forms of all irreducible representations $ρ: SB_3 \rightarrow GL_2(\mathbb{C})$ and we introduce the forms of some irreducible representations $ρ: SB_3 \rightarrow GL_3(\mathbb{C})$. The third finding is that we st…

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
Colin Defant, Melissa Sherman-Bennett, Nathan Williams
https://arxiv.org/abs/2509.11497 https://…

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences.…

Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings
John Guaschi (LMNO, UNICAEN, CNRS, NU), Daniel Juan-Pineda (CCM)
https://arxiv.org/abs/2509.02195

Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings
We describe the lower algebraic $K$-theory of the integral group ring of both the pure and full braid groups of the real projective plane $\mathbb{R}P^2$ with $3$ strings, as well as that of the integral group ring of the mapping class group of $\mathbb{R}P^2$ with $3$ marked points. In addition, we give a general formula for the algebraic $K$-theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $k \geq 3$.

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…

The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence
Daciberg Lima Gon\c{c}alves, John Guaschi, Carolina de Miranda e Pereiro
https://arxiv.org/abs/2509.02707

The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence
We study some generalisations to mixed braid groups of the Fadell-Neuwirth short exact sequence and the possible splitting of this sequence. In certain cases, we determine conditions under which the projection from the mixed braid group $B_{n_{1},\ldots,n_{k}}(M)$ to $B_{n_{1},\ldots, n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_{1}, \ldots, n_{k},q \in \mathbb{N}$, and $1\leq q \leq k-1$. For $k\geq 2$ and $q=k-1$, we show that this projection admits a s…

Crosslisted article(s) found for math.AG. https://arxiv.org/list/math.AG/new
[1/1]:
- Torsion in the Braid Monodromy of Elliptic Fibrations
Faye Jackson
https:/…

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.

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

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…

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

Monoidal categorification and quantum affine algebras III
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
https://arxiv.org/abs/2509.14552 https://a…

Monoidal categorification and quantum affine algebras III
Let $U_q'(\mathfrak{g})$ be an arbitrary quantum affine algebra of either untwisted or twisted type, and let $\mathscr{C}_{\mathfrak{g}}^0$ be its Hernandez-Leclerc category. We denote by $\mathsf{B}$ the braid group determined by the simply-laced finite type Lie algebra $ \mathsf{g}$ associated with $U_q'(\mathfrak{g})$. For any complete duality datum $\mathbb{D}$ and any sequence of simple roots of $\mathsf{g}$, we construct the corresponding affine cuspidal modules and affine determinantial …

Hurwitz space components; and the Coleman-Oort Conjecture
Michael D. Fried
https://arxiv.org/abs/2509.08904 https://arxiv.org/pdf/2509.08904

Hurwitz space components; and the Coleman-Oort Conjecture
Hurwitz spaces are moduli of isotopy classes of covers. A specific space is formed from a finite group G and C, r of its conjugacy classes and an equivalence relation \dagger. Components, interpret as a braid orbits on Nielsen classes. Fried-Völklein 1991 related absolute (corresponding to a permutation representation, T, of G) and inner equivalence classes. It noted two situations producing multiple components: #1. The action of a normalizer subgroup from $T$ on components; and #2. dist…

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…

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.

Torsion in the Braid Monodromy of Elliptic Fibrations
Faye Jackson
https://arxiv.org/abs/2509.14454 https://arxiv.org/pdf/2509.14454

Torsion in the Braid Monodromy of Elliptic Fibrations
Given an elliptic fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, let $\operatorname{Br}(π) < \operatorname{Mod}(S^2,Δ)$ be the subgroup of the spherical braid group consisting of those braids that lift to a fiber-preserving diffeomorphism of $M$. We classify the order $n = |Δ|$ elements of $\operatorname{Br}(π)$ up to conjugacy in $\operatorname{Br}(π)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety …

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

Determinant of the crossing matrix of a braid
Ayaka Shimizu
https://arxiv.org/abs/2509.08464 https://arxiv.org/pdf/2509.08464

Determinant of the crossing matrix of a braid
In this paper, we define a braid invariant, the purified determinant $P(b)$ of a braid $b$, considering the determinant of the crossing matrix, and show that $P(b_1 b_2)=P(b_2 b_1)$ for any pair of $n$-braids $b_1$ and $b_2$.

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

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 $G_2$.

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.

Crosslisted article(s) found for math.GR. https://arxiv.org/list/math.GR/new
[1/1]:
- Torsion in the Braid Monodromy of Elliptic Fibrations
Faye Jackson
https:/…

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 $\mathfrak{g}$.

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…

A large color $R$-matrix for $\mathfrak{sl}_3$
Angus Gruen, Lara San Mart\'in Su\'arez
https://arxiv.org/abs/2508.15171 https://arxiv.org/pdf/2508.…

A large color $R$-matrix for $\mathfrak{sl}_3$
We construct the invariant $F_K^{\mathfrak{sl}_3}\in\mathbb{Z}[q,q^{-1}][[x,y]]$ for any positive braid knot $K$, whose existence was conjectured by Park, building on earlier work of Gukov--Manolescu. The main step in our work extends a result by the first author on the invariant associated to symmetric representations of $\mathfrak{sl}_3$ to all irreducible representations. We conclude with a conjectural framework for constructing $F_K^{\mathfrak{sl}_N}$ for arbitrary $N$.