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Explicit diagonalization of the Markov form on theTemperley–Lieb algebra†

Published online by Cambridge University Press:  01 May 2007

JOSH GENAUER  and
NEAL W. STOLTZFUS
JOSH GENAUER
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail: [email protected]
NEAL W. STOLTZFUS
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, U.S.A. e-mail: [email protected]
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Abstract

In a fundamental paper in 1984, Vaughan Jones developed his new polynomialinvariant of knots using a Markov trace on the Temperley–Liebalgebra. Subsequently, Lickorish used the associated bilinear pairing toprovided an alternative proof for the existence of the 3-manifold invariants ofWitten, Reshetinkin and Turaev. A key property of this form is thenon-degeneracy of this form except at the parameter values±2cos(π/(n+1)) [7]. Koand Smolinsky derived a recursive formula for the determinants of specificminors of Markov's form, establishing the needed non-degeneracy[6]. In this paper,we define a triangular change of basis in which the form is diagonal andexplicitly compute the diagonal entries of this matrix as products of quotientsof Chebyshev polynomials, corroborating the determinant computation of Ko andSmolinsky. The method of proof employs a recursive method for defining therequired orthogonal basis elements in the Temperley–Lieb algebra,similar in spirit to Jones' and Wenzl's recursive formulafor a family of projectors in the Temperley–Lieb algebra. We define apartial order on the non-crossing chord diagram basis and give an explicitformula for a recursive construction of an orthogonal basis, via a recursionover this partial order. Finally we relate this orthogonal basis to basesconstructed using the calculus of trivalent graphs developed by Kauffman andLins [5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 469 - 485
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Blanchet, C., Habegger, N., Masbaum, G. and Vogel, P.. Three-manifold invariants derived from the Kauffman bracket. Topology 31, No. 4 (1992), 685–699.CrossRefGoogle Scholar
[2]Brualdi, R. A. . Introductory Combinatorics (North-Holland, 1977).Google Scholar
[3]Gilmer, P. M. and Harris, J. M.. On the Kauffman bracket skein module of the quaternionic manifold, arXiv:math.GT/0406152.Google Scholar
[4]Jones, V. F. R. . Index for subfactors. Invent. Math. 72 No. 1 (1983), 1–25.CrossRefGoogle Scholar
[5]Kauffman, L. H. and Lins, S. L..Temperley–Lieb recoupling theory and invariants of 3-manifolds. Ann. of Math. Stud. 134 (1994)Google Scholar
[6]Ko, K. H. and Smolinsky, L. S..A combinatorial matrix in 3-manifold theory.Pacific J. Math. 149, No. 2 (1991), 319–336.CrossRefGoogle Scholar
[7]Lickorish, W. B. R. . 3-Manifold invariants from the combinatorics of the Jones polynomial. Pacific J. Math. 149, No. 2 (1991), 337–347.Google Scholar
[8]Reshetikhin, N. Y. and Turaev, V. G.. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), 547–597.CrossRefGoogle Scholar
[9]Stanley, R. P. . Enumerative Combinatorics. vol. 2(Cambridge University Press, 1999), p. 581.CrossRefGoogle Scholar
[10]Witten, E. . Quantum field theory and Jones' polynomial. Comm. Math. Phys. 121 (1989), 351–399.CrossRefGoogle Scholar