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

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.