Vanishing of higher order Alexander‐type invariants of plane curves

José I. Cogolludo‐Agustín and Eva Elduque

Mathematische Nachrichten, 2023, vol. 296, issue 3, 1026-1040

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′$C^{\prime }$ and C′′$C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of C′$C^{\prime }$ and C′′$C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ΔCmulti$\Delta ^{\operatorname{multi}}_C$ is a power of (t−1)$(t-1)$, and we characterize when ΔCmulti=1$\Delta ^{\operatorname{multi}}_C=1$ in terms of the defining equations of C′$C^{\prime }$ and C′′$C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

Date: 2023
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