  Two continuous maps $f, g : \mathbb{C}^2\to\mathbb{C}^2$ are said to be
topologically equivalent if there exist homeomorphisms
$\varphi,\psi:\mathbb{C}^2\to\mathbb{C}^2$ satisfying $\psi\circ f\circ\varphi
= g$. It is known that there are finitely many topologically non-equivalent
polynomial maps $\mathbb{C}^2\to\mathbb{C}^2$ with any given degree $d$. The
number $T(d)$ of these topological types is known only whenever $d=2$. In this
paper, we describe the topology of generic complex polynomial maps on the plane
using the corresponding pair of Newton polytopes and establish a method for
constructing topologically non-equivalent maps of degree $d$. We furthermore
provide a software implementation of the resulting algorithm, and present lower
bounds on $T(d)$ whenever $d=3$ and $d=4$.
