A generalisation of simple Harnack curves

Abstract

In this paper, we suggest the following generalisation of Mikhalkin's simple Harnack curves: a generalised simple Harnack curve is a parametrised real algebraic curve in $(\mathbb{C}^*)^2$ with totally real logarithmic Gauss map. We investigate which of the many properties of simple Harnack curves survive the latter generalisation. We also show how tropical geometry allows to construct plenty of examples. Since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition where only real isolated double points can appear, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we give their topological classification as in \cite{Mikh} and show how such curves can be recovered from their spine.