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A toric braid is a braid whose closure is a torus link in ℝ 3 . Manturov [A combinatorial representation of links by quasitoric braids, European J. Combin.23(2) (2002) 207–212] generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization. [ABSTRACT FROM AUTHOR]

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as n → ∞ , the proportion of simple braids on n strands that have positive topological entropy tends toward 100%. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple three-strand braids that may be of independent interest. [ABSTRACT FROM AUTHOR]

Given a representation φ : B n → G n of the braid group B n , n ≥ 2 into a group G n , we are considering the problem of whether it is possible to extend this representation to a representation Φ : S M n → A n , where S M n is the singular braid monoid and A n is an associative algebra, in which the group of units contains G n . We also investigate the possibility of extending the representation Φ : S M n → A n to a representation Φ ~ : S B n → A n of the singular braid group S B n . On the other hand, given two linear representations φ 1 , φ 2 : H → G L m (k) of a group H into a general linear group over a field k , we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of S B n which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation. [ABSTRACT FROM AUTHOR]

We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups. [ABSTRACT FROM AUTHOR]

Let n ≥ 2. Let V B n (resp. V P n) denote the virtual braid group (resp. virtual pure braid group), let W B n (resp. W P n) denote the welded braid group (resp. welded pure braid group) and let U V B n (resp. U V P n) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for n ≥ 4 , the group V P n and for n ≥ 3 the groups W P n and U V P n are characteristic subgroups of V B n , W B n and U V B n , respectively. In the second part of the paper we show that, for n ≥ 2 , the virtual braid group V B n , the unrestricted virtual pure braid group U V P n , and the unrestricted virtual braid group U V B n have the R ∞ -property. As a consequence of the technique used for few strings we also prove that, for n = 2 , 3 , 4 , the welded braid group W B n has the R ∞ -property and that for n = 2 the corresponding pure braid groups have the R ∞ -property. On the other hand for n ≥ 3 it is unknown if the R ∞ -property holds or not for the virtual pure braid group V P n and the welded pure braid group W P n. [ABSTRACT FROM AUTHOR]

In this article, we show super-rigidity of Gromov’s random monster group. It is known from a paper of Assaf Naor and Lior Silberman that any homomorphic image of Gromov’s random monster group into a linear group is finite. It can be also derived from the previously known results that the same result is true for a- L p -menable groups and K-amenable groups. We extend these results and prove that any morphism ϕ α from Gromov’s random monster group Γ α to a countable discrete group G has finite image for almost all α , where G is any of the following types of groups: mapping class group M C G (S g , b) , braid group B n , outer automorphism group of a free group O u t (F N) , automorphism group of a free group A u t (F N) and hierarchically hyperbolic group. For acylindrically hyperbolic groups, we deduce that the homomorphic image is absolutely elliptic. We introduce another property called hereditary super-rigidity, which is the property of super-rigidity for all finite-index sub-groups. It immediately follows from the literature that Γ α has hereditary super-rigidity with respect to an a- L p -menable group or a K-amenable group for a.e. α . In this article, we establish a stability result for the groups with respect to which Γ α has super-rigidity and hereditary super-rigidity. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

We introduce the notion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch. Using (virtual) multi-switches we introduce a general approach of how to construct representations of (virtual) braid groups by automorphisms of algebraic systems. As a corollary we introduce new representations of virtual braid groups which generalize several previously known representations. [ABSTRACT FROM AUTHOR]

We develop a method based on the Burau matrix to detect conditions on the linking numbers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a 'subsymmetry' property; in particular at most two such braids can be mutually conjugate. As an addition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image. [ABSTRACT FROM AUTHOR]

Fix a suitable link on the disk. Recently, F. Morabito associates each Hamiltonian symplecticmorphism preserving the link to a braid type. Based on this construction, Morabito defines a family of pseudometrics on the braid groups by using the Hofer metric. In this paper, we show that two Hamiltonian symplecticmorphisms define the same braid type provided that their Hofer distance is sufficiently small. As a corollary, the pseudometrics defined by Morabito are nondegenerate. [ABSTRACT FROM AUTHOR]

We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line X does not depend on the parameters of X . Finally we prove that the determinant of the matrix obtained by taking the values of n Z -linear functions defined on the Grothendieck group K 0 (X) ≃ Z n of the elements of a full exceptional sequence is an invariant, up to sign. [ABSTRACT FROM AUTHOR]

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number p (K) of a knot K is the minimum number of loops among all petal diagrams of K. Let T n , s denote the (n , s) -torus knot for relatively prime integers 2 ≤ n < s. Recently, Kim et al. proved that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 whenever s ≡ ± 1 mod n. They conjectured that the inequality holds without the assumption s ≡ ± 1 mod n. They also showed that p (T n , s) = 2 s − 1 whenever 2 ≤ n < s < 2 n and n ≡ 1 mod s − n. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 holds for any 2 ≤ n < s. We also show that p (T n , s) = 2 s − 1 holds for any 2 ≤ n < s < 2 n. Our proofs construct petal grid diagrams for any torus knots. [ABSTRACT FROM AUTHOR]

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 has three natural extensions, namely the braid group B n , the twin group T n and the triplet group 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 and L n when n ≥ 4 . We also determine subquotients of principal congruence subgroups of T n , and identify the pure twin group P ⁢ T n and the pure triplet group P ⁢ L 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 / T n ′′ and L n / P ⁢ L n ′ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ . [ABSTRACT FROM AUTHOR]

We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface C of genus g ≥ 2 with C / G ≅ P 1. [ABSTRACT FROM AUTHOR]

In this paper, by introducing an isomorphism from the Mihailova subgroup of F 2 × F 2 to the Mihailova subgroups of a braid group, we give an explicit presentation of Mihailova subgroups of a braid group. Hence, in a braid group, there are some Mihailova subgroups experiencing unsolvable subgroup membership problem. Based on this, we propose a post-quantum signature scheme of the Wang–Hu scheme, and we show that the signature scheme is free of quantum computational attack. [ABSTRACT FROM AUTHOR]

Let Cn be the group of conjugating automorphisms. We study the representation ρ of Cn, an extension of Lawrence-Krammer representation of the braid group Bn. It is known that the representation ρ is unfaithful for n ≥ 5, the cases n = 3, 4 remain open. In our work, we make attempts towards the faithfulness of ρ in the case n = 3. [ABSTRACT FROM AUTHOR]

We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur–Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of Uq(glN). Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitly the irreducible representations and the branching rules. Finally, we aim to give an algebraic description of the centralisers of the tensor products of Uq(glN)-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of Uq(sl2). [ABSTRACT FROM AUTHOR]

Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variety by an explicit polynomial equation. The mixed braid group $ B_{m, n} $ is a subgroup of the Artin braid group $ B_{m+n} $. In this paper we find the ambiguity-free presentation and the Hilbert series of canonical words of mixed braid monoid $ M\!B_{2, 2} $.

It is known that the Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation u t - Δ u = | ∇ u | p with p > 2 , which has important applications in stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after the appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. The solutions eventually become classical again for large time, but in-between they may undergo losses and recoveries of boundary conditions at multiple times (as well as GBU at multiple times), hence forming a quite complex dynamics. In this paper, with the help of braid group theory combined with inner/outer region analysis, we give a complete classification of all viscosity solutions in one-dimensional case. Namely, we fully describe the (countably many, type II) rates and space–time profiles of gradient blow-up (GBU) or recovery of boundary condition (RBC), we characterize them at any time when such a phenomenon occurs, and we moreover determine whether the space–time profiles are stable or unstable. These results can be modified in radial domains in general dimensions. Even for type II blow-up in other PDEs, as far as we know, there has been no complete classification except Mizoguchi (Commun Pure Appl Math 75:1870–1886, 2022), in which the argument relies on features peculiar to chemotaxis system. Whereas there are many results on construction of special type II blow-up solutions of PDEs with investigation of stability/instability of bubble, determination of stability/instability of space–time profile for general solutions has not been done. A key in our proofs is to focus on braid group algebraic structure with respect to vanishing intersections with the singular steady state, as a GBU or RBC time is approached. In turn, the GBU and RBC rates and profiles, as well as their stability/instability, can be given a complete geometric characterization by the number of vanishing intersections. We construct special solutions in bounded and unbounded intervals in both GBU and RBC cases, based on methods from Herrero and Velázquez (Preprint, 1994), and then we apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space–time profile. In the process, careful construction of special solutions with specific behaviors in intermediate and outer regions, which is far from bubble and the RBC point, plays an essential role. The application of such techniques to viscosity solutions is completely new. [ABSTRACT FROM AUTHOR]

We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with ℝ2. This result is nontrivial already for unramified coverings, in which case the lifting problem is well-known to reduce to the purely algebraic problem of factoring the monodromy map to the symmetric group Sn through the braid group Bn. In our approach, this factorization is often achieved as a consequence of a stronger property: a factorization through a free group. In the reverse direction we show that any non-abelian surface group has infinitely many finite simple non-abelian groups quotients with characteristic kernels which do not contain any simple loop and hence the quotient maps do not factor through free groups. By a pullback construction, finite dimensional Hermitian representations of braid groups provide invariants for the braided surfaces. We show that the strong equivalence classes of braided surfaces are separated by such invariants if and only if they are profinitely separated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain representations of the Iwahori–Hecke algebra and a cabled version of the Temperley–Lieb representation.

In this paper, we compute the twisted Alexander invariant of the braid group associated with the Tong–Yang–Ma representation. [ABSTRACT FROM AUTHOR]

We study stability conditions on the Calabi–Yau- N categories associated to an affine type A n quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order N − 2. 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 A based on the Khovanov–Seidel–Thomas method. [ABSTRACT FROM AUTHOR]

Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variety by an explicit polynomial equation. The mixed braid group Bm,n is a subgroup of the Artin braid group Bm+n. In this paper we find the ambiguity-free presentation and the Hilbert series of canonical words of mixed braid monoid MB2,2. [ABSTRACT FROM AUTHOR]

In this paper, by introducing an isomorphism from the Mihailova subgroup of F2×F2 to the Mihailova subgroups of a braid group, we give an explicit presentation of Mihailova subgroups of a braid group. Hence, in a braid group, there are some Mihailova subgroups experiencing unsolvable subgroup membership problem. Based on this, we propose a post-quantum signature scheme of the Wang–Hu scheme, and we show that the signature scheme is free of quantum computational attack.

This paper in memory of Patrick Dehornoy (1952–2019) is an invitation to Garside theory for mainstream geometric group theorists interested in mapping class groups, curve complexes, and the geometry of Artin–Tits groups. [ABSTRACT FROM AUTHOR]

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type F ∞ . The proof utilized certain contractible cube complexes, which in this paper we prove are CAT (0) . We then use this fact to compute the geometric invariants Σ m (F br) of the pure braided Thompson group F br . Only the first invariant Σ 1 (F br) was previously known. A consequence of our computation is that as soon as a subgroup of F br containing the commutator subgroup [ F br , F br ] is finitely presented, it is automatically of type F ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

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 construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of K [ x ] / (x n) and the interval of rational permutation braids in the braid group on n strands. Hence, there are only finitely many tilting modules. [ABSTRACT FROM AUTHOR]

We consider the Albeverio-Rabanovich linear representation π of the braid group B3. After specializing the indeterminates used in defining the representation to non-zero complex numbers, we prove that its restriction to the pure braid group P3 of dimension three is irreducible.

We prove that the quantum toroidal algebras E s associated with different root systems s of gl m | n type are isomorphic. We also show the existence of Miki automorphism of E s , which exchanges the vertical and horizontal subalgebras. To obtain these results, we establish an action of the toroidal braid group on the direct sum ⊕ s E s of all such algebras. [ABSTRACT FROM AUTHOR]

Let C b n be the group of basis conjugating automorphisms of a free group F n , and C n the group of conjugating automorphisms of F n . Valerij G. Bardakov has constructed representations of C b n , C n in the groups G L n (Z [ t 1 ± 1 , ... , t n ± 1 ]) and in G L n (Z [ t ± 1 ]) respectively, where t 1 , ... , t n , t are indeterminate variables. We show that these representations are reducible and we determine the irreducible components of the representations in G L n (C) , which are obtained by giving values to the variables above. Next, we consider the tensor product of the representations of C b n , C n and study their irreduciblity in the case n = 3 . [ABSTRACT FROM AUTHOR]

Let n ≥ 3. In this paper, we show that for any abelian subgroup G of s n the crystallographic group B n / [ P n , P n ] has Bieberbach subgroups Γ G with holonomy group G. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group Γ G . As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of Γ G and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold Γ G with fundamental group the Bieberbach group Γ G ≤ B n / [ P n , P n ]. [ABSTRACT FROM AUTHOR]

An odd Coxeter group W is one which admits a Coxeter system (W, S) for which all the exponents m ij are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs V (W , S) are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete description of the automorphism group of this group, and derive a sufficient condition for the splitting of the automorphism group as a semi-direct product of the inner and the outer automorphism groups. As applications, we prove that Coxeter groups in this family satisfy the R ∞ -property and are (co)-Hopfian. We compare structural properties, automorphism groups, R ∞ -property and (co)-Hopfianity of a special odd Coxeter group whose only finite exponent is three with the braid group and the twin group. [ABSTRACT FROM AUTHOR]