2nd Order Algebraic Concordance and Twisted Blanchfield Forms

In my first year report [16] I described how to write down explicitly the chain complex C∗(X̃) for the universal cover of the knot exterior X, given a knot diagram. In this report we describe work to fit this chain complex into an algebraic group which we construct which measures a “2nd order slice-ness obstruction,” in the sense that it obstructs the concordance class of a knot lying in the level F1.5 of the Cochran-Orr-Teichner filtration of the knot concordance group C1 (see [3]). The obstruction is universal in that it does not depend on a choice of vanishing of first order obstructions. We also describe a project to calculate these obstructions explicitly using twisted Blanchfield forms: using certain metabelian representations of π1(X) which depend, as in the work of Casson-Gordon [1], on the vanishing of the Q/Z linking form on the 1st homology of a branched covering of the knot, it is possible to twist the coefficients in such a way as to make it computationally reasonable to compute the Blanchfield pairing from the symmetric chain complex of the zero framed surgery MK on the knot. We describe attempts to compute this for a certain knot, called 12631a in the tables, whose slice status is currently unknown.