Torres-type formulas for link signatures

Abstract

We investigate the limits of the multivariable signature function $\ddot{I}{ƒ}_L$ of a $\hat{I}¼$-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 $\ddot{I}{ƒ}_L$ by generalized Seifert surfaces and forms. The second relies on a new extension of $\ddot{I}{ƒ}_L$ from its usual domain $(S^1\setminus \{1\}{)}^{\hat{I}}¼$ to the full torus ${\mathbb{T}}^{\hat{I}}¼$ together with a Torres-type formula for $\ddot{I}{ƒ}_L$, 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 $1$.

52 pages, 9 figures. v5: we amended the construction of the extended signatures, as the construction in v4 relied on the incorrect Lemma 2.11. All the main results are unaffected