Abstract
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso–Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the “eigenvalue conjecture”, which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6×6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.
Highlights
Knot theory [1]-[5] is intimately connected to representation theory via the Reshetikhin-Turaev (RT) formalism [6]-[18]
The topologically invariant knot polynomials, which we are looking for, keep some memory about the difference between S and S, because their transformations from vertical to topological framings are different. This was the origin of additional factors in knot polynomials of the previous subsection (e.g. each R-matrix in the previous subsection has to be multiplied by t4 =4 in order to obtain the correct formulas (26), (27) from (23)-(25), and an additional factor4n in (19)) and this will produce some extra common powers of t = uvw in our formulas below, they are the only memory of orientation dependence which survives in the universal sector of knot theory
Though the adjoint representation is not symmetric, mutants could not be separated by our universal adjoint polynomials for arborescent knots, for the reason that we already mentioned in s.3.4: each representation appears in Adj⊗2 only once and fingers are not matrices
Summary
Knot theory [1]-[5] is intimately connected to representation theory via the Reshetikhin-Turaev (RT) formalism [6]-[18]. Note that in terms of representation theory, U is constructed as the Racah matrix that relates the two maps: (V ⊗ V ) ⊗V → W and V ⊗ (V ⊗ V ) → W (for the sake of simplicity, we discuss here only the case of knots, Q not links when one is not obliged to consider three coinciding representations V ) This matrix is involved in evaluating the knot invariants from 3-strand braid representation. What we do in the present paper, we solve two problems at once: we use the coincidence of these Racah matrices in order to apply the EC to U , using its symmetricity and the form of the first row This allows us to restore the matrix U and we use S = U to construct the universal adjoint polynomials of the arborescent knots. The second paper [42] contains the results for more general knots that can be presented by ”fingered 3-strand closed braids” [16, 17]
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