Topic
Braid
About: Braid is a research topic. Over the lifetime, 3617 publications have been published within this topic receiving 42328 citations. The topic is also known as: plait.
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TL;DR: A theory of braids leading to a classification was given in my paper "Theorie der Zopfe" in vol.
Abstract: A theory of braids leading to a classification was given in my paper "Theorie der Zopfe" in vol. 4 of the Hamburger Abhandlungen (quoted as Z). Most of the proofs are entirely intuitive. That of the main theorem in ?7 is not even convincing. It is possible to correct the proofs. The difficulties that one encounters if one tries to do so come from the fact that projection of the braid, which is an excellent tool for intuitive investigations, is a very clumsy one for rigorous proofs. This has lead me to abandon projections altogether. We shall use the more powerful tool of braid coordinates and obtain thereby farther reaching results of greater generality. A few words about the initial definitions. The fact that we assume of a braid string that it ends in a straight line is of course unimportant. It could be replaced by limit assumptions or introduction of infinite points. The present definition was selected because it makes some of the discussions easier and may be replaced any time by another one. I also wish to stress the fact that the definition of s-isotopy is of a provisional character only and is replaced later (Definition 3) by a general notion of isotopy. More than half of the paper is of a geometric nature. In this part we develop some results that may escape an intuitive investigation (Theorem 7 to 10). We do not prove (as has been done in Z) that the relations (18) (19) are defining relations for the braid group. We refer the reader to a paper by F. Bohnenblust1 where a proof of this fact and of many of our results is given by purely group theoretical methods. Later the proofs become more algebraic. With the developed tools we are able to give a unique normal form for every braid2 (Theorem 17, fig. 4 and remark following Theorem 18). In Theorem 19 we determine the center of the braid group and finally we give a characterisation of braids of braids. I would like to mention in this introduction a few of the more important of the unsolved problems: 1) Assume that two braids can be deformed into each other by a deformation of the most general nature including self intersection of each string but avoiding intersection of two different strings. Are they isotopic? One would be inclined to doubt it. Theorem 8 solves, however, a special case of this problem. 2) In Definition 3, we introduce a notion of isotopy that is already very general. What conditions must be put on a many to many mapping so that the result of Theorem 9 still holds?
1,116 citations
Journal Article•
TL;DR: The theory of braids as discussed by the authors shows the interplay of two disciplines of pure mathematics? topology, used in the definition of Braids, and the theory of groups used in their treatment.
Abstract: The theory of braids shows the interplay of two disciplines of pure mathematics? topology, used in the definition of braids, and the theory of groups, used in their treatment. The fundamentals of the theory can be understood without too much technical knowledge. It originated from a much older problem in pure mathematics?the classification of knots. Much progress has been achieved in this field ; but all the prog ress seems only to emphasize the extreme difficulty of the problem. Today we are still very far from a complete solution. In view of this fact it is advisable to study objects that are in some fashion similar to knots, yet simple enough so as to make a complete classification possible. Braids are such objects. In order to develop the theory of braids we first explain what we call a weaving pat tern of order ( being an ordinary inte gral number which is taken to be 5 in Fig. 1). Let Li and L2 be two parallel straight lines in space with given orientation in the same sense (indicated by arrows). If is a point on Lh Q a point on L2, we shall sometimes join and Q by a curve c. In our drawings we can only indicate the pro jection of c onto the plane containing L\
785 citations
31 Jan 1975
516 citations
20 Aug 2000
TL;DR: The aim of this article is to show that the braid groups can serve as a good source to enrich cryptography and to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in thebraid groups.
Abstract: The braid groups are infinite non-commutative groups naturally arising from geometric braids The aim of this article is twofold One is to show that the braid groups can serve as a good source to enrich cryptography The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers (ii) The group operations can be performed efficiently (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups The efficiency of our systems is demonstrated by their speed and information rate The security of our systems is based on topological, combinatorial and group-theoretical problems that are intractible according to our current mathematical knowledge The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design
504 citations
TL;DR: In this paper, an 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.
Abstract: 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 ∆ 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.
341 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 141 |
| 2024 | 166 |
| 2023 | 199 |
| 2022 | 441 |
| 2021 | 113 |
| 2020 | 135 |