One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.
This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations.
We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass from a_k to the empty word is quadratic with respect to the length of a_k.
A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a sigma-definite word representative that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.
We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $σ_1,\dots,σ_{n-1}$. Our study addresses invariance properties of these presentations and the presented groups under various operations on the words. In particular we prove that the group does only depend on the corresponding element in the positive braid monoid, and under mild hypotheses only on its conjugacy class in the braid group.
Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. It is well-known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braids on surfaces, providing the first algorithm (to our knowledge) which solves the word problem for braid groups on surfaces with boundary in polynomial time and space. In the case of surfaces without boundary, braid combing needs to use a section from the fundamental group of the …
We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible set of "half-braid" relations to the ordinary braid relations produces a spanning set: in the untwisted case, this occurs for the Coxeter systems which are finite with rank two or type $A_n$, or affine with rank three or type $\tilde A_n$. These results generalize recent work of Hu and Zhang on the finite classical cases.
In this paper, we obtain Groebner-Shirshov (non-commutative Gröbner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word" $δ$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form in the braid groups, and thus a new algorithm for solving the word problem in these groups.
In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word prob…
In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.
Let $Γ$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,Σ$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and let $E_Γ$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a simplicial complex $Ω(Γ)$ having the same homotopy type as $E_Γ$. We observe that, if $T \subset S$, then $Ω(Γ_T)$ naturally embeds into $Ω(Γ)$. We prove that this embedding admits a retraction $π_T: Ω(Γ) \to Ω(Γ_T)$, and we deduce several to…
In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.V.Chaynikov on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators is described as well.
We found Groebner-Shirshov basis for the braid semigroup $B^+_{n+1}$. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group.
We study braid diagrams with a minimal number of crossings. Such braid diagrams correspond to geodesic words for the braid groups with standard Artin generators. We prove that a diagram of a homogeneous braid is minimal if and only if it is homogeneous. We conjecture that monoids of homogeneous braids are Artin-Tits monoids and prove that monoids of alternating braids are right-angled Artin monoids. Using this, we give a lower bound on the growth rate of the braid groups.
We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the $n$-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in \cite{FH} can be verified. We discover more characteristics of graph braid groups: the $n$-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and …
We describe a new algorithm which for each braid returns a quasi-geodesic sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears either only positively or only negatively.
We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. …
Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.
We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round.
We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$…
In this paper, we shall introduce two monoids. One is called a PM-monoid which contains the symmetric group, the other is called a braid PM-monoid which contains the braid group. We shall develop the theory of PM-monoids and that of braid PM-monoids. The PM-monoids is obtained in the context of the compactification of projective linear group defined by Mutsumi Saito. The structure of PM-monoids is described in terms of matched pairs. We can define braid PM-monoid using a presentation for the PM-monoid. As main results, we show that braid PM-monoids are described by geometric braids and we find…
A conjecture of Dehornoy claims that, given a presentation of an Artin-Tits group, every word that represents the identity can be transformed into the trivial word using the braid relations, together with certain rules (between pairs of words that are not both positive) that can be derived directly from the braid relations, as well as free reduction, but without introducing trivial factors $ss^{-1} $ or $s^{-1} s$. This conjecture is known to be true for Artin-Tits groups of spherical type or of FC type. We prove the conjecture for Artin--Tits groups of sufficiently large type.
We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number of strands.
In the last decade, a number of public key cryptosystems based on com- binatorial group theoretic problems in braid groups have been proposed. We survey these cryptosystems and some known attacks on them. This survey includes: Basic facts on braid groups and on the Garside normal form of its elements, some known algorithms for solving the word problem in the braid group, the major public-key cryptosystems based on the braid group, and some of the known attacks on these cryptosystems. We conclude with a discussion of future directions (which includes also a description of cryptosystems which ar…
We establish strong constraints on the kernel of the (reduced) Burau representation $β_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $β_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $β_4\left(g\right)$, in terms of the Garside normal form of $g\in B_4$ and a new product decomposition of positive braids. We develop cance…
Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such as the super summit set and the ultra summit set. Their complexities are proportional to the size of the finite set, even when there exist very short conjugating elements. However, ultra summit sets are very large in some cases especially for reducible braids and periodic braids. Some possible approaches to resolve this difficulty would be either to use dif…
We define geodesic normal forms for the general series of complex reflection groups 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) over the generating set of the presentation of Corran-Picantin. Using these geodesic normal forms, we construct intervals in G(e,e,n) that are 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) and get a complete classification. For the other Garside groups that appea…
We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clarifies some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions. After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances. Our main concern is Artin's braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We sho…
