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Topological quantum computation
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations ..."
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Cited by 118 (16 self)
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Abstract. The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in WittenChernSimons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2Dmagnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e−αℓ, where ℓ is a length scale, and α is some positive constant. In contrast, the “presumptive ” qubitmodel of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about 10−4) before computation can be stabilized. Quantum computation is a catchall for several models of computation based on a theoretical ability to manufacture, manipulate and measure quantum states. In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19],
New Publickey Cryptosystem Using Braid Groups
 Advances in cryptology—CRYPTO 2000 (Santa Barbara, CA), 166–183, Lecture Notes in Comput. Sci. 1880
, 2000
"... Abstract. The braid groups are infinite noncommutative 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 ..."
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Cited by 106 (4 self)
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Abstract. The braid groups are infinite noncommutative 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 grouptheoretical 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. Key words: public key cryptosystem, braid group, conjugacy problem, key exchange, hard problem, noncommutative group, oneway function, public key infrastructure 1
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 103 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Braid Groups are Linear
 J. Amer. Math. Soc
, 2001
"... The braid group Bn can be dened as the mapping class group of the npunctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over R. Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case n = ..."
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Cited by 83 (7 self)
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The braid group Bn can be dened as the mapping class group of the npunctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over R. Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case n = 4. In this paper, we show that it is faithful for all n. 1.
Simulation of topological field theories by quantum computers
 Comm.Math.Phys.227
"... Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has ..."
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Cited by 81 (15 self)
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Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is twofold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.
Infinitesimal presentations of the Torelli groups
 Ha2] [HL] [KM] [Jo83
, 1997
"... 2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608 ..."
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Cited by 63 (6 self)
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2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 61 (9 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
Vassiliev Homotopy String Link Invariants
, 1995
"... . We investigate Vassiliev homotopy invariants of string links, and nd that in this particular case, most of the questions left unanswered in [3] can be answered armatively. In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants co ..."
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Cited by 58 (4 self)
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. We investigate Vassiliev homotopy invariants of string links, and nd that in this particular case, most of the questions left unanswered in [3] can be answered armatively. In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants come from marked surfaces as in [3], using the same construction that in the case of knots gives the HOMFLY and Kauman polynomials. Alongside, the Milnor invariants of string links are shown to be Vassiliev invariants, and it is reproven, by elementary means, that Vassiliev invariants classify braids. Contents 1. Introduction 1 2. Vassiliev invariants of string links 3 2.1. A brief review of [3] 3 2.2. Vassiliev invariants of string links 4 3. Vassiliev homotopy string link invariants 5 4. Vassiliev invariants classify braids 9 4.1. Braids 9 4.2. Braids with double points 11 5. On the Milnor invariants 13 5.1. Vassiliev invariants classify string links up to homotopy 13 5.2. The Milnor ...
Compact Stein surfaces with boundary as branched covers of B4, Invent
 Math
"... We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3manifold is Stein fillable iff it has a positive openbook decompo ..."
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Cited by 48 (4 self)
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We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3manifold is Stein fillable iff it has a positive openbook decomposition.