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 $(S^1)^{\mu}$ 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 $1$.