Tootfinder

Random braids and random walks on finite groups
Heorhii Zhylinskyi
https://arxiv.org/abs/2507.21999 https://arxiv.org/pdf/2507.21999

Random braids and random walks on finite groups
In this paper, we study properties of random walks on finite groups and later use them to obtain the limiting braid length expectation and component number of braid closure in a model of random braids, which is constructed by lifting elements of random walk on a Coxeter group to a braid group.

Earlier today I took a break form work and came out of the office, only to find my wife in full Braids, Cabal Minion cosplay in the living room.
Still kinda weirded-out. She's not _that_ into M:tG, so from my perspective it truly came out of left field. But it also kinda looks good on her, which says what about me exactly?

Genus Zero Kashiwara-Vergne Solutions from Braids
Zsuzsanna Dancso, Iva Halacheva, Guillaume Laplante-Anfossi, Marcy Robertson, Chandan Singh
https://arxiv.org/abs/2507.16243

Genus Zero Kashiwara-Vergne Solutions from Braids
Using the language of moperads-monoids in the category of right modules over an operad-we reinterpret the Alekseev-Enriquez-Torossian construction of Kashiwara-Vergne (KV) solutions from associators. We show that any isomorphism between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck-Teichmüller module groups act on…

Tropical Ptolemy Transformations and Invariants of Braids
Gurnoor Singh
https://arxiv.org/abs/2506.12085 https://arxiv.org/pdf/2506.1…

Tropical Ptolemy Transformations and Invariants of Braids
It often happens in mathematics that one and the same equation is known under different names in different areas of mathematics. The famous pentagon identity appears in low-dimensional topology in different ways. In this paper, we use the tropical version of the Ptolemy equation to construct invariants of braids.

Bonded braids and the Markov theorem
Paolo Cavicchioli, Bo\v{s}tjan Gabrov\v{s}ek, Matic Simoni\v{c}
https://arxiv.org/abs/2507.04565 https://

Bonded braids and the Markov theorem
Bonded knots arise naturally in topological protein modeling, where intramolecular interactions such as disulfide bridges stabilize folded configurations. These structures extend classical knot theory by incorporating embedded graphs, and have been formalized as bonded knots. In this paper, we develop the algebraic theory of bonded braids, introducing the bonded braid monoid in the topological and rigid settings, which encodes both classical braid crossings and (rigid) bonded connections. We pr…

Comparing cluster algebras on braid varieties
Roger Casals, Pavel Galashin, Mikhail Gorsky, Linhui Shen, Melissa Sherman-Bennett, Jos\'e Simental
https://arxiv.org/abs/2508.03816

Comparing cluster algebras on braid varieties
Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster alge…

Pointed Racks and Their Applications to Braid Theory
Angel Apollos, Jose Ceniceros
https://arxiv.org/abs/2508.03013 https://arxiv.org/pdf/2508.03013…

Pointed Racks and Their Applications to Braid Theory
We define a new algebraic structure called a \emph{pointed rack} and use it to construct ambient isotopy invariants of $ n $-braids. We first introduce an integer-valued invariant of braids using pointed racks. This is then strengthened by defining a matrix-valued invariant using racks. Moreover, our invariant determines the rack coloring invariant previously defined for the closure of the braid. Finally, we include examples of braids that are distinguished by these new invariants.

🇺🇦 #NowPlaying on #KEXP's #VarietyMix
The Braids:
🎵 Bohemian Rhapsody
#TheBraids
https://francolippi.bandcamp.com/track/the-braids-bohemian-rhapsody-franco-lippi-extended-reggaeboot-mix
https://open.spotify.com/track/3TAWWWIwm0g7PDGGL70tNn

This https://arxiv.org/abs/2303.07133 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Braid stability for periodic orbits of area-preserving surface diffeomorphisms
We consider an area-preserving diffeomorphism of a compact surface, which is assumed to be an irrational rotation near each boundary component. A finite set of periodic orbits of the diffeomorphism gives rise to a braid in the mapping torus. We show that under some nondegeneracy hypotheses, the isotopy classes of braids that arise from finite sets of periodic orbits are stable under Hamiltonian perturbations that are small with respect to the Hofer metric. A corollary is that for a Hamiltonian …

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…

This https://arxiv.org/abs/2505.22403 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Knot invariants from representations of braids by automorphisms of a free group
We describe an alternative way of computing Alexander polynomials of knots/links, based on the Artin representation of the corresponding braids by automorphisms of a free group. Then we apply the same method to other representations of braid groups discovered by Wada and compare the corresponding isotopic invariants to Alexander polynomials.

This https://arxiv.org/abs/2502.02939 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On grid homology for diagonal knots
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. We compare diagonal knots to various classes of knots, such as positive braids, fibered positive knots, and $L$-space knots.