Vanishing of higher order Alexander‐type invariants of plane curves

Abstract

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve 𝐶 given as a transversal union of plane curves 𝐶 ′ and 𝐶 ′′ in terms of the finiteness and the vanishing properties of the invariants of 𝐶 ′ and 𝐶 ′′ , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial Δ multi 𝐶 is a power of (𝑡 − 1), and we characterize when Δ multi 𝐶 = 1 in terms of the defining equations of 𝐶 ′ and 𝐶 ′′ . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.