Signatures of real algebraic curves via plumbing diagrams

TL;DR

This paper introduces new invariants for real algebraic curves in the projective plane, using plumbing diagrams, and applies these to obstruct certain schemes from being realizable as algebraic curves.

Contribution

It develops formulas for signature and nullity invariants, and extends the Murasugi-Tristram inequality to real algebraic curves, providing new tools for their classification.

Findings

Derived formulas for Casson-Gordon invariants of graph manifolds

Established obstructions for realizability of schemes as algebraic curves

Extended signature invariants to complex schemes in the real projective plane

Abstract

We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and signatures of graph links.