A new presentation of the n-string braid group B n is studied. Using it, a new solution to the word problem in B n is 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. Academic Press
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.
Abstract. We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knotsNaik's and Choi-Ko-Song's improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots -but are not equivariantly slice.
Algorithmic solutions to the conjugacy problem in the braid groups B n , n=2, 3, 4, ... were given in earlier work. This note concerns the computation of two integer class invariants, known as ''inf'' and ''sup.'' A key issue in both algorithms is the number m of times one must ''cycle'' (resp. ''decycle'') in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that m is bounded above by ((n 2 − n)/2) − 1 in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford 45, 479-497) and by n − 2 in the situation stated by authors (1998, Adv. Math. 139, 322-353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length.
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 [10] 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 the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in [9] and so we propose a similar conjecture for higher braid indices.2010 Mathematics Subject Classification. Primary 20F36, 20F65, 57M15.
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