Universal weight systems and the Melvin-Morton expansion of the colored Jones knot invariant (Q1356323)

In [Commun. Math. Phys. 169, No. 3, 501-520 (1995; Zbl 0845.57007)]\ \textit{P. M. Melvin} and \textit{H. R. Morton} studied properties of the `coloured Jones function' of a knot \(K\). This is the quantum \(Sl2\) invariant of \(K\) coloured by the irreducible module of dimension \(k\), when regarded as a power series in the quantum variable \(h\) with coefficients depending on \(k\). They conjectured that the Alexander polynomial of \(K\) could be retrieved from the coloured Jones function, and proposed an explicit formula to determine it. The conjecture was subsequently proved in a number of different ways, initially in a heuristic approach by \textit{L. Rozansky} [ibid. 175, No. 2, 275-296 (1996; Zbl 0872.57010), 297-318 (1996; Zbl 0872.57011)], followed by the direct proof of \textit{D. Bar-Natan} and \textit{S. Garoufalidis} [Invent. Math. 125, No. 1, 103-133 (1996; Zbl 0855.57004)], and more recent accounts by \textit{A. Kricker, B. Spence} and \textit{I. Aitchison} [Cabling the Vassiliev invariant, J. Knot Theory Ramifications 6, No. 3, 327-358 (1997)]\ and by \textit{S. Chmutov} [ibid. 7, No. 1, 23-40 (1998; Zbl 0892.57002)]; see also work of \textit{B. I. Kurpita} and \textit{K. Murasugi} [A graphical approach to the Melvin-Morton conjecture I, Topology Appl. 82, No. 1-3, 297-316 (1998)] and of \textit{X. S. Lin} and \textit{Z. Wang} [Random walk on knot diagrams, colored Jones polynomial and Ihara-Selberg zeta function, E-print math. GT/9812039]. It coincides with an increased awareness of the widespread appearance of Vassiliev invariants and of the techniques available to handle them. In this paper the author uses the universal weight system of the Lie algebra of \(Sl2\), which assigns to every chord diagram an element of the centre of the universal enveloping algebra of \(Sl2\). For \(Sl2\) the centre is generated by a single element \(c\), the quadratic Casimir, and each chord diagram then determines a polynomial in \(c\) whose highest degree terms can be readily identified. Any invariant coming from a representation of \(Sl2\) then determines a weight system whose evaluation on any diagram can be found by simply evaluating the polynomial at the value of \(c\) determined by the representation. This fact, together with a systematic method of handling the weight systems for unframed knots, then provides suitable information about the relevant terms in the coloured Jones weight system. The remainder of the proof relies on combining this with properties of the Alexander weight system, and invoking the results about canonical invariants which allow explicit relations between the knot invariants to be deduced from relations established on the weight systems.