Explicit diagonalization of the Markov form on the Temperley–Lieb algebra

Abstract

AbstractIn a fundamental paper in 1984, Vaughan Jones developed his new polynomial invariant of knots using a Markov trace on the Temperley–Lieb algebra. Subsequently, Lickorish used the associated bilinear pairing to provided an alternative proof for the existence of the 3-manifold invariants of Witten, Reshetinkin and Turaev. A key property of this form is the non-degeneracy of this form except at the parameter values ±2cos(π/(n+1)) [7]. Ko and Smolinsky derived a recursive formula for the determinants of specific minors 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 and explicitly compute the diagonal entries of this matrix as products of quotients of Chebyshev polynomials, corroborating the determinant computation of Ko and Smolinsky. The method of proof employs a recursive method for defining the required orthogonal basis elements in the Temperley–Lieb algebra, similar in spirit to Jones' and Wenzl's recursive formula for a family of projectors in the Temperley–Lieb algebra. We define a partial order on the non-crossing chord diagram basis and give an explicit formula for a recursive construction of an orthogonal basis, via a recursion over this partial order. Finally we relate this orthogonal basis to bases constructed using the calculus of trivalent graphs developed by Kauffman and Lins [5].