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Solving the Braid Word Problem Via the Fundamental Group
Abstract
The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological point of view, and thus using a new computerized representation of some elements of the fundamental group we gave a solution for its word problem. In this paper we will give an algorithm that will make it possible to transform the new presentation into a syntactic presentation. This will make it possible to computerize the group operation to sets of elements of the fundamental group, which are isomorphic to the braid group. More over we will show that it is sufficient enough to look at the syntactic presentation in order to solve the braid word problem, resulting with a better and faster braid word solution.
In this paper we present the Braid Monodromy Type (BMT) of curves and surfaces. The BMT can distinguish between non-isotopic curves; between different families of surfaces of general type; between connected components of moduli space of surfaces finer than Sieberg-Witten invariants; and between symplectic 4-manifolds.
A method for constructing algorithms solving the word and comparison problems for mapping class groups (in particular, for the braid group) is presented, and a family of one-side invariant orderings on the mapping class group of a surface with boundary is described. A method for constructing comparison algorithms for all finite orderings on the mapping class group of any surface with boundary is described, a fast and simple comparison algorithm for the Dehornoy order on the braid group is presented, examples of normal forms for braid groups are given, and algorithms for finding the forms are indicated. Bibliography: 15 titles.
Wir untersuchen Darstellungsattacken auf das Zopf-Diffie-Hellman Schlüsselaustauschprotokoll via Lawrence-Krammer- und Burau-Darstellung der Zopfgruppe. Dabei entwickeln wir eine verbesserte Inversionsheuristik für die Burau-Darstellung in linearer Zeit. Desweiteren führen wir ein verallgemeinertes Anshel-Anshel-Goldfeld-Verfahren für Magmen und ein verbessertes Fiat-Shamir-artiges Authentifizierungsprotokoll für LD-Systeme ein.
Es ist bekannt, dass die Monodromie der Milnor-Faserung einer isolierten Singularität quasiunipotent ist. Dies ist nicht länger der Fall, wenn man eine nicht-lokale Monodromie um mehrere Singularitäten betrachtet. Wir studieren hier den Fall von Familien von (endlich vielen) Morse-Singularitäten. Für den Fall, dass eine solche Familie eine Morsifikation einer isolierten Singularität ist, zeigen wir, dass sämtliche Monodromien, die zu einfachen Schleifen um eine Teilfamilie der zugehörigen kritischen Punkte gehören, schon dann quasiunipotent sind, wenn dies stets für Schleifen um nur zwei kritische Punkte gilt. Wir stellen die Vermutung auf, dass dies auch (aus rein kombinatorischen Gründen) im allgemeinen Fall gilt und beweisen eine abgeschwächte Form dieser Vermutung. It is a well-known fact that the monodromy of the Milnor fibration of an isolated singularity is quasiunipotent. This holds no longer true if a non-local monodromy around several singularities is considered. Here the case of families of (finitely many) Morse singularities will be studied. For the case that such a family arises from a morsification of an isolated singularity it will be proven that all monodromies corresponding to simple loops around a subfamily of the corresponding critical values are already quasiunipotent if and only if this is always the case for simple loops around only two critical values. We conjecture that this is (for purely combinatorial reasons) also true for the general case and prove a weaker analogon of this conjecture.
In this paper we present the Braid Monodromy Type (BMT) of curves and surfaces; past, present and future. The BMT is an invariant that can distinguish between non-isotopic curves; between different families of surfaces of general type; between connected components of moduli space of surfaces and between non symplectmorphic 4-manifolds. BMT is a finer invariant than the Sieberg-Witten invariants. Consider 2 simply connected surfaces of general type with the same Chern classes. It is known that if they are in the same deformation class, they are diffeomorphic to each other. Are there computable invariants distinguishing between these 2 classes? The new invariant, proposed here, is located between the 2 classes. In this paper we shall introduce the new invariant, state the current results and pose an open question.
A new presentation of then-string braid groupBnis studied. Using it, a new solution to the word problem inBnis obtained which retains most of the desirable features of the Garside–Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.
We give an easily handled algorithm for the word problem in each of Artin's braid groups, Bn, based on Garside's methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined in terms of positive braids, and apply this to compare braids with different powers ¢r of the fundamental half-twist braid ¢. This leads to an improvement of Garside's conjugacy algorithm, using a much smaller finite subset of each conjugacy class, which we term the super summit set, to represent the class, in place of Garside's summit set.
A new presentation for the 4-braid group (called the band-generator presentation) is introduced. The word problem, the conjugacy problem and the shortest word problem for this presentation are solved.
At the beginning of the 1930s, as a means of studying knots, E. Artin introduced a concept of a (mathematical) braid(s). This remarkable insight itself was not sufficient to sustain research in this area, and so it slowly began to wither. However, in the 1950s this concept of braids was found to have applications in other fields, and this gave fresh impetus to the study of braids, rekindling research in this area. The iridescent hue of this concept flowering into full bloom and activity occurred in 1984, when V. Jones put into action with inordinate success the original aim of Artin, i.e., the application of braids to knot theory. In this chapter our intention is to introduce certain necessary aspects of the theory of braids that will prove useful when we explain recent developments in knot theory in the subsequent chapters.
This text presents some recent developments in left distributive algebra induced by set theory and their applications to the combinatorics of braids mainly in terms of order properties and existence of special decompositions.
We give a new algorithm to solve with cubic complexity the word problem in the braid group Bn. This allows us to give an effective answer to the conjugacy problem in B4,B5.
We describe a new method for comparing braid words which relies both on the automatic structure of the braid groups and on the existence of a linear ordering on braids. This syntactical algorithm is a direct generalization of the classical word reduction used in the description of free groups, and is more efficient in practice than all previously known methods. We consider in this paper the classical braid isotopy problem, i.e., the question of deciding if a given two-dimensional diagram made of a series of mutually crossing strands can be transformed into another one by moving strands but not allowing one to pass through another one. As is well-known, this problem became a question of algebra after E. Artin in the 20's has rephrased it as the word problem for a family of effectively presented groups, Artin's braid groups B n . Many solutions have been described, beginning with Artin's original construction that uses the geometric idea of combing the braids to obtain a normal form for braid words and a decomposition of the groups B n as semidirect products of free groups ([1]). The starting point for modern braid comparison method is the purely algebraic result by Gar-side [10] that every braid can be decomposed into a quotient of two positive braids, i.e., of braids where all crossings have the same orientation. Several algorithms have been con-structed: [10] itself, then [17] (cf. [9]), [8], [6], [16], [15]. These methods take advantage of the special form of the relations in the standard presentation of the groups B n , mainly in terms of the geometry of the associated Cayley graph. In particular the existence of a (bi)automatic structure on B n guarantees the existence of a quadratic isoperimetric inequality, and explains the efficiency of the practical algorithms deduced from this ap-proach: they have a polynomial complexity with respect to the length of the braid words, even a quadratic complexity when the number of strands is fixed.
A new presentation for the 4-braid group (called the band-generator presentation) is introduced. The word problem, the conjugacy problem and the shortest word problem for this presentation are solved.
One of the most interesting questions about a group is whether 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 geometric one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition in order to give a new approach for solving its word problem. Our algorithm, although not better in complexity, is faster in comparison with known algorithms for short braid words, 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.