다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

/pdf/quantum-field-theory-and-the-jones-polynomial-5897tp4ud9.pdf

Quantum field theory and the Jones polynomial

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Non-Abelian Anyons and Topological Quantum Computation

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {it Non-Abelian anyons}, meaning that they obey {it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the nu=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators for Bn. The second example shows that the correspondence of (b, n) with b is many-to-one, and a theorem of A. Markov [15] answers, in theory, the question of when two braids represent the same link. A Markov move of type 1 is the replacement of (6, n) by (gbg~, n) for any element g in Bn, and a Markov move of type 2 is the replacement of (6, n) by (6s J 1 , n-hl). Markov's theorem asserts that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups. Unforunately, although the conjugacy problem has been solved by F. Garside [8] within each braid group, there is no known algorithm to decide when (6, n) and (c, m) are equivalent. For a proof of Markov's theorem see J. Birman's book [4]. The difficulty of applying Markov's theorem has made it difficult to use braids to study links. The main evidence that they might be useful was the existence of a representation of dimension n — 1 of Bn discovered by W. Burau in [5]. The representation has a parameter t, and it turns out that the determinant of 1-(Burau matrix) gives the Alexander polynomial of the closed braid. Even so, the Alexander polynomial occurs with a normalization which seemed difficult to predict.

/pdf/a-polynomial-invariant-for-knots-via-von-neumann-algebras-2w7tukqxnv.pdf

A polynomial invariant for knots via von Neumann algebras

Index for subfactors

By studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. It is expressed using a trace, discovered by Ocneanu, on the Hecke algebras of type A. A certain specialization of the polynomial, whose discovery predated and inspired the two-variable one, is seen to come in two inequivalent ways, from a Hecke algebra quotient and a linear functional on it which has already been used in statistical mechanics. The two-variable polynomial was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste, and Przytycki-Traczyk.

Hecke algebra representations of braid groups and link polynomials

On utilise trois sortes differentes de modeles de mecanique statistique pour construire des invariants de nœuds. Les modeles de sommets emergent comme les plus generaux

/pdf/on-knot-invariants-related-to-some-statistical-mechanical-rcvkmiwl81.pdf

On knot invariants related to some statistical mechanical models

We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II_1 factor, and vice versa.

Vaughan F. R. Jones

Papers

Index for subfactors

A polynomial invariant for knots via von Neumann algebras

Hecke algebra representations of braid groups and link polynomials

On knot invariants related to some statistical mechanical models

Planar algebras, I