Spectrum of plane curves via knot theory

In this paper, we use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram–Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by Steenbrink and Varchenko on semicontinuity of the spectrum of singular points under deformations and result of Némethi and Sabbah on semicontinuity of the spectrum at infinity regarding families of polynomial maps. We also relate the spectrum at infinity of a polynomial map with the collection of the spectra of singular points of a chosen fiber.