Vassiliev invariants and a strange identity related to the Dedekind eta-function

Abstract

In this paper the "function" F(q)=∑n=0∞(1-q)(1-q2)(1-q n) is studied. The series does not converge in any open set, but has well-defined values and derivatives of all orders when q is a root of unity. It is shown that the coefficients of its Taylor expansion at q=1 are equal to the numbers ξD of "regular linearized chord diagrams" as defined by Stoimenow and hence give an upper bound (the best currently known) for the number of linearly independent Vassiliev invariants of degree D. There are similar expansions at other roots of unity. The same values and derivatives of all orders at all roots of unity are obtained as the limiting value of the function -12∑n∈Z(-1)n|6n+1|q(3n2+n)/2, the "derivative of order one-half" of the Dedekind eta-function, and also exhibit a kind of modular behavior which can be seen as an example of a generalization of the classical theory of periods of modular forms to the case of half-integral weight. Functions of a similar type also occurred in recent joint work with Lawrence in connection with the Witten-Reshetikhin-Turaev invariants of knots. © 2001 Elsevier Science Ltd.