For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\lbrace 0,1\rbrace$ and a set of points which is dense in $(32/27,\infty )$. Sokal proved that the corresponding set of complex zeros is dense in $\mathbb C$.

I am interested whether there are analogous results for Tutte polynomials and their zeros. In particular, is it known whether the union of all curves in $\mathbb C^2$ cut out by the Tutte polynomials of graphs is dense in $\mathbb C^2$? Pointers towards the behavior of real zeros are also welcome!