In 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].