Smooth tropical complete intersection curves of genus 3 in $$\mathbb {R}^3$$

Abstract

We develop a method for describing the tropical complete intersection of a tropical hypersurface and a tropical plane in \(\mathbb {R}^3\). This involves a method for determining the topological type of the intersection of a tropical plane curve and \(\mathbb {R}_{\le 0}^2\) by using a polyhedral complex. As an application, we study smooth tropical complete intersection curves of genus 3 in \(\mathbb {R}^3\). In particular, we show that there are no smooth tropical complete intersection curves in \(\mathbb {R}^3\) whose skeletons are the lollipop graph of genus 3. This gives a partial answer to a problem of Morrison in [6].