Torres-Type Formulas for Link Signatures

We investigate the limits of the multivariable signature function σL of a μ-component link L as some variable tends to 1 via two different approaches, three-dimensional and four-dimensional. The first uses the definition of σL by generalized Seifert surfaces and forms. The second relies on a new extension of σL from its usual domain (S1∖{1})μ to the full torus Tμ together with a Torres-type formula for σL, results which are of independent interest. Among several consequences, we obtain new estimates on the value of the Levine–Tristram signature of a link close to 1.