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The submonoid of the 3-strand braid group B 3 generated by σ 1 and σ 1 σ 2 is known to yield an exotic Garside structure on B 3 . We introduce and study an infinite family ( M n ) n ≥ 1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M 2 is isomorphic to the above monoid. The corresponding Garside group G ( M n ) is isomorphic to the ( n , n + 1 ) -torus knot group–which is isomorphic to B 3 for n = 2 and to the braid group of the exceptional complex reflection group G 12 for n = 3 . This yields a new Garside structure on ( n , n + 1 ) -torus knot groups, which already admit several distinct Garside structures. The ( n , n + 1 ) -torus knot group is an extension of B n + 1 , and the Garside monoid M n surjects onto the submonoid Σ n of B n + 1 generated by σ 1 , σ 1 σ 2 , … , σ 1 σ 2 ⋯ σ n , which is not a Garside monoid when n > 2 . Using a new presentation of B n + 1 that is similar to the presentation of G ( M n ) , we nevertheless check that Σ n is an Ore monoid with group of fractions isomorphic to B n + 1 , and give a conjectural presentation of it, similar to the defining presentation of M n . This partially answers a question of Dehornoy–Digne–Godelle–Krammer–Michel.

We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.

We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. We also investigate the proper quotients of it that appear in the realm of quantum groups, and describe another maximal quotient related to the usual Hecke algebras. Finally, we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by Vogel. We prove that these two are isomorphic for n ≤ 5 .

Degenerate solutions to the Yang-Baxter equation are studied by means of associated semibraces and groups. Using a characterization in terms of cycle sets, a non-degenerate part is separated from a purely degenerate one. More precisely, any cycle set X has a distinguished non-degenerate sub-cycle set X × . If X × is trivial, the permutation group G ( X ) of X admits a natural partial order, with a strong semibrace S ( X ) as its negative cone, so that G ( X ) is a group of left fractions of S ( X ) . The semibrace S ( X ) is constructed exactly like the self-similar closure of an L-algebra. It is proved that each non-trivial Garside group (e.g., Artin's braid group) gives rise to a degenerate cycle set. With a graded algebra related to the first Weyl algebra, a negative answer to a recent problem of Bonatto et al. (2021) [2] is obtained.

Effects connected with the mathematical theory of knots1 emerge in many areas of science, from physics2,3 to biology4. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems5, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.

For a semisimple complex Lie algebra g , the BGG category O is of particular interest in representation theory. It is known that Irving's shuffling functors Sh w , indexed by elements w ∈ W of the Weyl group, induce an action of the braid group B W associated to W on the derived categories D b ( O λ ) of blocks of O . We show that for maximal parabolic subalgebras p of sl n corresponding to the parabolic subgroup W p = S n − 1 × S 1 of S n , the derived shuffling functors L Sh s i are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives P p ( w ) are spherical objects, and the associated twist functors are naturally isomorphic to L Sh w [ 1 ] as auto-equivalences of D b ( O p ) . We give an overview of the main properties of the BGG category O , the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of O and the module categories of certain path algebras.

We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham Denham. We also use our result to compute the stable integral homology of the Milnor fiber.

Using the ordered analogue of Farley–Sabalka’s discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber’s equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.

We study the dual $\textrm{G}^{\ast }$ of a standard semisimple Poisson–Lie group $\textrm{G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ can be naturally embedded into a quotient algebra of a cluster Poisson algebra with a Weyl group action. The coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ admits a natural basis, which has positive integer structure coefficients and satisfies an invariance property under a braid group action. We continue the study of the moduli space $\mathscr{P}_{\textrm{G},{{\mathbb{S}}}}$ of $\textrm{G}$-local systems introduced in [ 16] and prove that the coordinate ring of $\mathscr{P}_{\textrm{G}, {{\mathbb{S}}}}$ coincides with its underlying cluster Poisson algebra.

We present an inspection of the statistics of particles including composite fermions on a torus starting from a braid group analysis. For this purpose we considered a system of electrons confined to the surface of a torus under the influence of a strong magnetic field and interacting through a general rotational invariant potential. An explanation of the appearance of the cyclotron braids as an effect of restriction imposed by magnetic field on braid trajectories which in analyzed case reduces the full braid group to one of its subgroups (i.e. cyclotron subgroups), is given. The modified Feynman path-integral method is also reproduced with some minor enhancements. We improve known results concerning on braid groups on a torus in two directions: we obtain new estimates in terms of cyclotron braid subgroups and cyclotron band generator, respectively; we demonstrate that only multi-loop generators are accessible in the fractional quantum regime well and we also formally explain the unique statistic of composite fermions by construct trial wave function for the system on a torus, based on this idea. The topological oddness of torus geometry can be driven by shifting of electrons between the two different group of generators allowed for an explanation in satisfactory accordance the both compact commensurability condition and some numerical calculations in toroidal geometry. Besides, our approach may shed some new light on few interesting aspects in better understanding the fractional quantum Hall effect.

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $\beta_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $\beta_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $\beta_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes., Comment: Mostly minor revisions in v2. The biggest change is that we now have a complete proof that the isomorphism on homology in the gas regime is induced by the inclusion map

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{\Gamma_l(P_n)\}_{l\in N}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$). We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/\Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/\Gamma_3(P_{|G|})$), where $|G|$ denotes the order of $G$. The result in the case of $B_{|G|}/\Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $Z_{p^r} \rtimes_{\theta} Z_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p \geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/\Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/\Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/\Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/\Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $Z_9 \rtimes Z_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.

We consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.

We consider the Birman-Hilden inclusion $\varphi\colon\mathfrak{Br}_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\varphi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\varphi^*(H_1(\Sigma_{g,1}))$ has only $4$-torsion., Comment: 17 pages, 3 figures, slight change in the introduction

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules., 25 pages, lots of figures, comments welcome

In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes., Clarify the main conjecture as a twisted version of dimension reduction. To appear in Comm. Math Phys

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g.\ fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell-Jones conjecture, the coarse Baum-Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a mean of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex., Comment: Small modifications according to the referee report, updated references. Final accepted version

We show that most of the genus-zero subgroups of the braid group $\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned: there is a very restricted class of \enquote{primitive} genus-zero subgroups such that these subgroups and their genus-zero intersections determine all the Alexander invariants. Then, we classify the primitive subgroups in a special subclass. This result implies the known classification of the dihedral covers of irreducible trigonal curves., Comment: 21 pages, 1 Table

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations., Comment: Compared to previous version the introduction and the List of references are extended, few remarks added

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is $\mathfrak{sl}_n$. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations., Substantial revision following referee comments. Section 2 has been shortened considerably. Lemmas 3.10 to 3.15 have been condensed into a single proof of Theorem 3.8. Sections 3.2 and 3.3 are now Sections 4 and 5, respectively. Removed Propositions 3.22 to 3.26 and replaced with Section 4.2. The Appendix has been split into three parts for readability. To appear in Journal of Algebra;38 pages

We analyze security properties of a two-party key-agreement protocol recently proposed by I. Anshel, D. Atkins, D. Goldfeld, and P. Gunnels, called Kayawood protocol. At the core of the protocol is an action (called E-multiplication) of a braid group on some finite set. The protocol assigns a secret element of a braid group to each party (private key). To disguise those elements, the protocol uses a so-called cloaking method that multiplies private keys on the left and on the right by specially designed elements (stabilizers for E-multiplication). We present a heuristic algorithm that allows a passive eavesdropper to recover Alice’s private key by removing cloaking elements. Our attack has 100% success rate on randomly generated instances of the protocol for the originally proposed parameter values and for recent proposals that suggest to insert many cloaking elements at random positions of the private key. Implementation of the attack is available on GitHub.

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level number. We then find braid descriptions for $(1,1)$-positions of all $2$-bridge knots providing upper bounds for their level numbers, and also show that the $(-2, 3, 7)$-pretzel knot has level number two., Comment: 24 pages, 17 figures

For any simply-laced GIM Lie algebra ${\mathscr{L}}$ , we present the definition of quantum universal enveloping algebra $U_{q}({\mathscr{L}})$ , and prove that there is a quantum universal enveloping algebra $U_{q}(\mathcal {A})$ of an associated Kac-Moody algebra $\mathcal {A}$ , together with an involution ( $\mathbb {Q}$ -linear) σ, such that $U_{q}({\mathscr{L}})$ is isomorphic to the $\mathbb {Q}(q)$ -extension $\widetilde {S}_{q}$ of the σ-involutory subalgebra Sq of $U_{q}(\mathcal {A})$ . This result gives a quantum version of Berman’s work (Berman Comm. Algebra 17, 3165–3185, 1989) in the simply-laced cases. Finally, we describe an automorphism group of $U_{q}({\mathscr{L}})$ consisting of Lusztig symmetries as a braid group.

We prove several conjectures about the cohomology of Deligne–Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broue–Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broue's abelian defect group conjecture and the specific form Broue conjectured for finite groups of Lie type [4, Reves 1 et 2] .

We construct bundles $$E_k({{\mathscr {A}}},{{\mathscr {F}}}) \rightarrow M$$ over the complement M of a complex hyperplane arrangement $${{\mathscr {A}}}$$ , depending on an integer $$k \geqslant 1$$ and a set $${{\mathscr {F}}}=\{f_1, \ldots , f_\mu \}$$ of continuous functions $$f_i :M \rightarrow {{\mathbb {C}}}$$ whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence–Krammer–Bigelow representation of the pure braid group. We display such families $${{\mathscr {F}}}$$ for rank two arrangements, reflection arrangements of types $$A_\ell $$ , $$B_\ell $$ , $$D_\ell $$ , $$F_4$$ , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group $$\pi _1(M)$$ . We describe the kernel of the product $$\rho _{{\mathscr {X}}}:G \rightarrow \prod _{S \in {{\mathscr {X}}}} G_S$$ of homomorphisms of a finitely-generated group G onto quotient groups $$G_S$$ determined by a family $${{\mathscr {X}}}$$ of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections $$G \rightarrow G_S$$ split in a compatible way, we show the image of $$\rho _{{\mathscr {X}}}$$ is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of $$\pi _1(M)$$ , including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina–Brady groups, and a dichotomy for a decomposable arrangement $${{\mathscr {A}}}$$ : either $$\pi _1(M)$$ has a conjugation-free presentation or it is not residually nilpotent.

We give a conjectural classification of virtually cocompactly cubulated Artin–Tits groups (i.e., having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin–Tits groups of spherical type, FC type, or two-dimensional type. A particular case is that for $n \geqslant 4$, the $n$-strand braid group is not virtually cocompactly cubulated.

We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges., 49 pages, 19 figures. Mild revision for clarity, figures added. Accepted for publication in Geometry & Topology, may vary slightly from published version

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group., Comment: 7 pages, final version; to appear in Geometriae Dedicata

Recently the third named author defined a 2-parametric family of groups $$G_{n}^{k}$$ [12]. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups $$G_{n}^{k}$$ and dynamical systems led to the discovery of the following fundamental principle: ‘‘If dynamical systems describing the motion of $$n$$ particles possess a nice codimension one property governed by exactly $$k$$ particles, then these dynamical systems admit a topological invariant valued in $$G_{n}^{k}$$ ’’. The $$G_{n}^{k}$$ groups have connections to different algebraic structures, Coxeter groups and Kirillov–Fomin algebras, to name just a few. Study of the $$G_{n}^{k}$$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. In the present paper we prove that word and conjugacy problems for certain $$G_{k+1}^{k}$$ groups are algorithmically solvable.

We introduce bijections between generalized type An noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley–Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley–Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on n + 1 strands. We then show that our bijections come from triangular base changes between the diagram basis and the various bases indexed by noncrossing partitions, by explicitly describing the orders giving triangularity. These orders were introduced in a joint paper with Williams and provide exotic lattice structures on noncrossing partitions. Several combinatorial objects are introduced along the way, including an involution on the set of noncrossing partitions.

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3$-sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set., 10 pages, 8 figures; minor changes from previous version

It is known that if there exists a Gr?bner-Shirshov basis for a group G, then we say that one of the decision problem, namely the word problem, is solvable for G as well. Therefore, as the main target of this paper, we will present a (non-commutative) Gr?bner-Shirshov basis for the braid group associated with the congruence classes of complex reflection group G12 which will give us normal forms of the elements of G12 and so will obtain a new algorithm to solve the word problem over it.

In a topological description of elementary matter proposed by Bilson-Thompson, the leptons and quarks of a single generation, together with the electroweak gauge bosons, are represented as elements of the framed braid group of three ribbons. By identifying these braids with emergent topological excitations of ribbon networks, it has been possible to encode this braid model into the framework of quantum geometry provided by loop quantum gravity. One major hurdle in this promising approach to unifying matter and space-time has been the difficulty in implementing any dynamics that reflects observed particle interactions. In the case of trivalent networks, it has not been possible to generate particle interactions, because the braids correspond to noiseless subsystems, meaning they commute with the evolution algebra generated by the local Pachner moves. In the case of tetravalent networks, interactions are only possible when the model's original simplicity, in which interactions take place via the composition of braids, is sacrificed. The main result of the present paper is to demonstrate that it is possible to preserve both the original classification of fermions, as well as their interaction via the braid product, if we embed the braid in a trivalent scheme and supplement the local Pachner moves, with a nonlocal and graph-changing one-handle attachment. Moreover, we use Kauffman-Lins recoupling theory to obtain invariants of braided networks that distinguish topological configurations associated to particles in the Bilson-Thompson model.

Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group $\Gamma_{G}$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of $\Gamma_{G}$ and determine the existence of Anosov diffeomorphisms and K\"ahler geometry of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group $\Gamma_{G}\leq B_n/[P_n,P_n]$., Comment: 16 pages

Artin’s braid group $$B_n$$ is generated by $$\sigma _1, \ldots , \sigma _{n-1}$$ subject to the relations $$\begin{aligned} \sigma _i \sigma _{i+1} \sigma _i = \sigma _{i+1} \sigma _i \sigma _{i+1}, \quad \sigma _i\sigma _j = \sigma _j \sigma _i \text { if } |i-j|>1. \end{aligned}$$ For complex parameters $$q_1,q_2$$ such that $$q_1q_2 \ne 0$$ , the group $$B_n$$ acts on the vector space $$\mathbf {E}= \sum _i \mathbb {C}\mathbf {e}_i$$ with basis $$\mathbf {e}_1, \ldots , \mathbf {e}_n$$ by $$\begin{aligned} \sigma _i \cdot \mathbf {e}_i= & {} (q_1+q_2)\mathbf {e}_i + q_1\mathbf {e}_{i+1}, \quad \sigma _i \cdot \mathbf {e}_{i+1} = -q_2\mathbf {e}_i, \\ \sigma _i \cdot \mathbf {e}_j= & {} q_1 \mathbf {e}_j \quad \text { if }\; j \ne i,i+1. \end{aligned}$$ This representation is (a slight generalization of) the Burau representation. If $$q = -q_2/q_1$$ is not a root of unity, we show that the algebra of all endomorphisms of $$\mathbf {E}^{\otimes r}$$ commuting with the $$B_n$$ -action is generated by the place-permutation action of the symmetric group $$S_r$$ and the operator $$p_1$$ , given by $$\begin{aligned} p_1(\mathbf {e}_{j_1} \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r}) = q^{j_1-1} \, \sum _{i=1}^n \mathbf {e}_i \otimes \mathbf {e}_{j_2} \otimes \cdots \otimes \mathbf {e}_{j_r} . \end{aligned}$$ Equivalently, as a $$(\mathbb {C}B_n, \mathcal {P}'_r([n]_q))$$ -bimodule, $$\mathbf {E}^{\otimes r}$$ satisfies Schur–Weyl duality, where $$\mathcal {P}'_r([n]_q)$$ is a certain subalgebra of the partition algebra $$\mathcal {P}_r([n]_q)$$ on 2r nodes with parameter $$[n]_q = 1+q+\cdots + q^{n-1}$$ , isomorphic to the semigroup algebra of the “rook monoid” studied by W. D. Munn, L. Solomon, and others.

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem)., Comment: 25 pages

Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear $q$-difference equations for $V_n$-valued meromorphic functions on a complex $n$-torus, with $V_n$ a module over the GL${}_n$-type extended affine Hecke algebra $\mathcal{H}_n$. The family $(\mathcal{H}_n)_{n\geq 0}$ of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms $\mathcal{H}_n\rightarrow\mathcal{H}_{n+1}$ the Hecke algebra descends of arc insertion at the affine braid group level. In this paper we consider qKZ towers $(f^{(n)})_{n\geq 0}$ of solutions, which consist of twisted-symmetric polynomial solutions $f^{(n)}$ ($n\geq 0$) of the qKZ equations that are compatible with the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. The compatibility is encoded by so-called braid recursion relations: $f^{(n+1)}(z_1,\ldots,z_{n},0)$ is required to coincide up to a quasi-constant factor with the push-forward of $f^{(n)}(z_1,\ldots,z_{n})$ by an intertwiner $\mu_{n}: V_{n}\rightarrow V_{n+1}$ of $\mathcal{H}_{n}$-modules, where $V_{n+1}$ is considered as an $\mathcal{H}_{n}$-module through the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower $(f^{(n)})_{n\geq 0}$ of solutions. The solutions $f^{(n)}$ are constructed from specialised dual non-symmetric Macdonald polynomials with specialised parameters using the Cherednik-Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, $f^{(n)}$ coincides with the (suitably normalized) ground state of the inhomogeneous dense $O(1)$ loop model on the half-infinite cylinder with circumference $n$., Comment: 45 pages. v2: main theorem (Thm. 4.7) strengthened. v3: minor typos corrected. To appear in Ann. Henri Poincare

We define geodesic normal forms for the general series of complex reflection groups G ( e , e , n ) G(e,e,n) . This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G ( e , e , n ) G(e,e,n) over the generating set of the presentation of Corran–Picantin. Using these geodesic normal forms, we construct intervals in G ( e , e , n ) G(e,e,n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B ( e , e , n ) B(e,e,n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.

We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf-Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.

We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group [Formula: see text] and we show that if [Formula: see text] satisfies these two conditions, then [Formula: see text] has a Dehornoy structure. Then, we show that the Artin groups of type [Formula: see text] and of type [Formula: see text], [Formula: see text] satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type [Formula: see text] coincides with the Dehornoy order.

We describe a class of irreducible non-equivalent unitary representations of the braid group B 3 in every dimension n ≥ 6 which depends continuously on n 2 / 6 + 1 real parameters. We show that the upper bound on the number of the parameters of which the class of irreducible non-equivalent unitary representations of B 3 depends smoothly is equal to n 2 / 4 + 2 . The proof is achieved by a construction of such a class. We also prove that the tensor product of the Burau unitarisable representation of B 4 and the irreducible unitary representation of B 4 that coincide on commuting standard generators always forms irreducible unitary representations for the braid group B 4 . This gives a new class of unitary representations for the braid group B 4 in 3n dimensions.

We prove that if a right-angled Artin group $A_\Gamma$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_\Gamma$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_\Gamma$ and $A_\Delta$ are commensurable and $\Gamma$ has no separating $k$-cliques for any $k\le n$ then neither does $\Delta$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the braid group $B_n$ is not abstractly commensurable to any group that splits non-trivially over a "free group-free" subgroup, and the same holds for $n\ge 3$ for the loop braid group $LB_n$. Our approach makes heavy use of the Bieri--Neumann--Strebel invariant., Comment: 14 pages

We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all three-braids. We also obtain effective upper and lower bounds for the entropy of pure three-braids in these terms. The proof leads to the study of the extremal length of classes of curves representing elements of the fundamental group of the twice punctured complex plane., Comment: 56 pages, 7 figures