Slopes, colored links and Kojima's $\eta$ concordance invariant

In this talk we will introduce an invariant, the slope, for a colored link in a homology sphere together with a suitable multiplicative character defined on the link group. The slope takes values in the complex number union infinity and it is real for finite order characters. It is a generalization of Kojima $\eta$-invariants and can be expressed as a quotient of Conway polynomials. It is also related to the correction term in Wall's non-additivity formula for the signatures of 4-manifolds, and as such it appears naturally as a correction term in the expression of the signature formula for the splice of two colored links. This is work in progress with Alex Degtyarev and Vincent Florens.