Torres-type formulas for link signatures

We investigate the limits of the multivariable signature function ${\sigma }_L$ of a $\mu$-component link $L$ as some variable tends to $1$ via two different approaches: a three-dimensional and a four-dimensional one. The first uses the definition of ${\sigma }_L$ by generalized Seifert surfaces and forms. The second relies on a new extension of ${\sigma }_L$ from its usual domain $(S^1\setminus \{1\}{)}^{\mu }$ to the full torus ${\mathbb{T}}^{\mu }$ together with a Torres-type formula for ${\sigma }_L$, 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 $1$.