  We define the slope of a colored link in an integral homology sphere,
associated to admissible characters on the link group. Away from a certain
singular locus, the slope is a rational function which can be regarded as a
multivariate generalization of the Kojima--Yamasaki $\eta$-function. It is the
ratio of two Conway potentials, provided that the latter makes sense;
otherwise, it is a new invariant. The slope is responsible for an extra
correction term in the signature formula for the splice of two links, in the
previously open exceptional case where both characters are admissible. Using a
similar construction for a special class of tangles, we formulate generalized
skein relations for the signature.
