A combinatorial matrix in 3-manifold theory

Ko, Ki Hyoung; Smolinsky, Lawrence

doi:10.2140/pjm.1991.149.319

Cited by 40 publications

(40 citation statements)

References 2 publications

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“…Jones for correspondence, to H. Wenzl for very helpful conversation, and to K. H. Ko and L. Smolinsky for their result [9] that confirmed the conviction that the general methods of this paper would work. Lickofish linear algebra.…”

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confidence: 63%

Three-manifolds and the Temperley-Lieb algebra

Lickorish

1991

Math. Ann.

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An array of invariants for closed 3-manifolds and for links in 3-manifolds has been revealed by Witten [17] using the inspiration of quantum field theory. When the 3-manifold is the 3-sphere, these link invariants are essentially the Jones polynomial (or one of its generalisations) of the link, evaluated at various complex roots of unity. A proof of the existence of such invariants has been given by . Building on Kirby's theorem [7] concerning the different ways of obtaining a 3-manifold via surgery on the 3-sphere, they use deep results from the theory of quantum groups and the representation theory of Lie algebras. This paper gives an alternative approach, based on only the general outline of their method. The result obtained here establishes those new invariants that, in other interpretations, correspond to the Lie group SU(2). This proof of the invariants' existence, which also starts with Kirby's theorem, uses Kauffman's (easy) bracket invariant [-5] of regular isotopy classes of planar link diagrams. The behaviour of this invariant at roots of unity is explored using the discipline of the Temperley-Lieb algebra I-1, 6] (that is also used in statistical mechanics in the calculation of the partition function of the Ports model), and the results are blended with some of the elementary tricks of linear skein theory [11]. The results needed (and here proved) from the Temperley-Lieb algebra are implicit in work of Jones [4] which appeared before the advent of his Jones polynomial (see also [3]). Of course, the Kauffman bracket is but a clever reformulation of the Jones polynomial. The nature of the 3-manifold invariants is described in a fairly simple way, but calculations are by no means easy and will not be attempted here. Some of these calculations have been performed and discussed by Kirby and Melvin 18] using some of the few (sixteen) roots of unity at which the Jones polynomial can be expressed in terms of more classical invariants; they do at least show that the invariants are not trivial.The paper is, apart from its use of Kirby's surgery theorem, intended to be entirely elementary and self-contained. Much of it is but an exercise in elementary

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confidence: 63%

Three-manifolds and the Temperley-Lieb algebra

Lickorish

1991

Math. Ann.

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“…The matrix M q which has rows and columns indexed by chord diagrams and has entries (M q ) C 1 ,C 2 = C 1 , C 2 q is known as the Gram matrix of the Temperley-Lieb algebra (also called the meander matrix). Jones [Jon83] determined the values of q for which M q is positive definite and positive semidefinite, Ko and Smolinsky [KS91] determined when M q is nonsingular, and Di Francesco, Golinelli, and Guitter [DFGG97] gave an explicit diagonalization of M q . For our results on groves we shall use the fact that lim q→0 M q /q is nonsingular [DFGG97, Eqn (5.18)], and for our results on the double-dimer model we shall use the fact that M 2 is nonsingular [DFGG97, Eqn (5.6)].…”

Section: Grove Partitionsmentioning

confidence: 99%

Boundary partitions in trees and dimers

Kenyon¹,

Wilson²

2010

Trans. Amer. Math. Soc.

118

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Abstract. Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure; i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and, as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have a direct application to connection probabilities for multiple-strand SLE 2 , SLE 8 , and SLE 4 .

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“…Since then it has from time to time been studied by mathematicians in various contexts such as the folding of a strip of stamps [3,4] or folding of maps [5]. More recently it has been related to enumerations of ovals in planar algebraic curves [6] and the classification of 3-manifolds [7]. During the last decade or so has it has received considerable attention in other areas of science.…”

Section: Introductionmentioning

confidence: 99%

A transfer matrix approach to the enumeration of plane meanders

Jensen

2000

J. Phys. A: Math. Gen.

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A closed plane meander of order n is a closed self-avoiding curve intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.

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A combinatorial matrix in 3-manifold theory

Cited by 40 publications

References 2 publications

Three-manifolds and the Temperley-Lieb algebra

Three-manifolds and the Temperley-Lieb algebra

Boundary partitions in trees and dimers

A transfer matrix approach to the enumeration of plane meanders

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