Braid Group Research Articles

Surface Houghton groups

AbstractFor every $$n\ge 2$$ n ≥ 2 , the surface Houghton group$${\mathcal {B}}_n$$ B n is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups $${\mathcal {B}}_n$$ B n are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $${\mathcal {B}}_n$$ B n . As countable mapping class groups of infinite type surfaces, the groups $$\mathcal {B}_n$$ B n lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $$\mathcal {B}_n$$ B n is of type $$\text {F}_{n-1}$$ F n - 1 , but not of type $$\text {FP}_{n}$$ FP n , analogous to the braided Houghton groups.

A new Garside structure on torus knot groups and some complex braid groups

Several distinct Garside monoids having torus knot groups as groups of fractions are known. For [Formula: see text] two coprime integers, we introduce a new Garside monoid [Formula: see text] having as Garside group the [Formula: see text]-torus knot group, thereby generalizing to all torus knot groups a construction that we previously gave for the [Formula: see text]-torus knot group. As a byproduct, we obtain new Garside structures for the braid groups of a few exceptional complex reflection groups of rank two. Analogous Garside structures are also constructed for a few additional braid groups of exceptional complex reflection groups of rank two which are not isomorphic to torus knot groups, namely, for [Formula: see text] and for dihedral Artin groups of even type.

Deformed Cartan matrices and generalized preprojective algebras II: general type

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß–Leclerc–Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura–Pestun in their study of quiver mathcal {W}-algebras.

Congruence subgroups and crystallographic quotients of small Coxeter groups

Abstract Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n {S_{n}} has three natural extensions, namely the braid group B n {B_{n}} , the twin group T n {T_{n}} and the triplet group L n {L_{n}} . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n {T_{n}} and L n {L_{n}} when n ≥ 4 {n\geq 4} . We also determine subquotients of principal congruence subgroups of T n {T_{n}} , and identify the pure twin group P ⁢ T n {PT_{n}} and the pure triplet group P ⁢ L n {PL_{n}} with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ {T_{n}/PT_{n}^{\prime}} , T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} and L n / P ⁢ L n ′ {L_{n}/PL_{n}^{\prime}} are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} .

Braid group action and quasi-split affine 𝚤quantum groups I

This is the first of our papers on quasi-split affine quantum symmetric pairs ( U ~ ( g ^ ) , U ~ ı ) \big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big ) , focusing on the real rank one case, i.e., g = s l 3 \mathfrak g = \mathfrak {sl}_3 equipped with a diagram involution. We construct explicitly a relative braid group action of type A 2 ( 2 ) A_2^{(2)} on the affine ı \imath quantum group U ~ ı \widetilde {{\mathbf U}}^\imath . Real and imaginary root vectors for U ~ ı \widetilde {{\mathbf U}}^\imath are constructed, and a Drinfeld type presentation of U ~ ı \widetilde {{\mathbf U}}^\imath is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine ı \imath quantum groups in the sequels.

The Long–Moody construction and twisted Alexander invariants

In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its matrix presentation is written using the Fox derivative, and also a relation with twisted Alexander invariants. Communicated by Eric Jespers

TORIC REFLECTION GROUPS

Abstract Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $k\geq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J-groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra36(6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

Quasi-homomorphisms of quantum cluster algebras

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser.For a quantum Grassmannian cluster algebra Cq[Gr(k,n)], we show that there is an associated braid group and each generator σi of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that σi also preserves r-term (r≥4) quantum Plücker relations of Cq[Gr(k,n)] and other relations which cannot be derived from quantum Plücker relations (if any). Up to this conjecture, we show that σi is a quasi-automorphism of Cq[Gr(k,n)] and the braid group acts on Cq[Gr(k,n)].

Pauli topological subsystem codes from Abelian anyon theories

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the Z4(1) anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.

The artin braid group actions on the set of spin structures on a surface

The artin braid group actions on the set of spin structures on a surface

Differential operator realization of braid group action on ıquantum groups

Differential operator realization of braid group action on <i>ı</i>quantum groups

An intrinsic approach to relative braid group symmetries on ı$\imath$quantum groups

AbstractWe initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.

The evaluation of bilateral stenting using braided or laser-cut self-expandable metallic stent for malignant hilar biliary obstruction.

Bilateral self-expandable metallic stent (SEMS) placement for unresectable malignant hilar biliary obstruction (UMHBO) is an effective option for biliary drainage with long-term stent patency. Laser-cut and braided SEMS can be used for bilateral SEMS placement. This study aimed to clarify any differences in the clinical features and proper use of the laser-cut and braided SEMS placement using the stent-in-stent method for UMHBO. In this study, 78 patients who underwent bilateral stent-in-stent SEMS placement for UMHBO were included. The patients were divided into the laser-cut (n = 33) and braided groups (n = 45). Both groups were compared for technical and clinical success, adverse events (AEs), time to recurrent biliary obstruction (TRBO), overall survival, and endoscopic reintervention (ERI). There were no significant differences in technical and clinical success rates (laser-cut vs. braided group, 97% vs. 95.6%, P = 1.0), AEs (21.2% vs. 15.6%. P = 0.56), median TRBO (242days vs. 140days, P = 0.36), and median overall survival (654days vs. 675days, P = 0.58). ERI was required in 15 patients in the laser-cut group and in 20 patients in the braided group. The technical and clinical success rates of ERI (60% vs. 85%) were not significantly different (P = 0.13); however, the median ERI procedure time was significantly longer in the laser-cut group (38min) than in the braided group (22min; P = 0.02). No significant difference in initial SEMS placement was noted between the laser-cut and braided groups; however, the laser-cut group required a longer ERI procedure time than that required by the braided group. The use of braided SEMS may be a convenient option for ERI.

The ZX-calculus as a language for topological quantum computation

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.

Gain-loss-induced non-Abelian Bloch braids

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups B2 and B3 of two and three bands, respectively. While B2 is trivially Abelian, the group B3 features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing.

Solutions of the Yang–Baxter–like matrix equation with 3×3 diagonalizable coefficient matrix

The Yang–Baxter–like matrix equation plays an important role in quantum group theory, knot theory and braid groups, which has received great attention from physicists and mathematicians. A difficult and important open problem is to find all the solutions of the Yang–Baxter–like matrix equation. As a matter of fact, it is difficult to find all of the solutions even when the coefficient matrix is a 3 × 3 matrix. To the best of our knowledge, when the coefficient matrix is a 3 × 3 diagonalizable complex matrix with three distinct nonzero eigenvalues, finding all the solutions of the Yang–Baxter–like matrix equation is still an open problem. In order to fill-in this gap, we first present a sufficient and necessary condition for the commuting solutions of the Yang–Baxter–like matrix equation, and give all the commuting solutions of the matrix equation. Second, with the help of a simplified matrix equation, we derive all the non–commuting solutions of the Yang–Baxter–like matrix equation by discussing whether the off–diagonal elements of the solutions are zero or not.

Graph of groups decompositions of graph braid groups

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a nontrivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.

Near braces and p$p$‐deformed braided groups

AbstractMotivated by recent findings on the derivation of parametric noninvolutive solutions of the Yang–Baxter equation, we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi‐parametric, nondegenerate, noninvolutive solutions of the set‐theoretic Yang–Baxter equation. These solutions are generalizations of the known ones coming from braces and skew braces. Bijective maps associated to the inverse solutions are also constructed. Furthermore, we introduce the generalized notion of ‐deformed braided groups and ‐braidings and we show that every ‐braiding is a solution of the braid equation. We also show that certain multi‐parametric maps within the near braces provide special cases of ‐braidings.

Smallest noncyclic quotients of braid and mapping class groups

Smallest noncyclic quotients of braid and mapping class groups