Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

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!

edited Jul 30 at 6:07

asked Jul 30 at 4:57

Gjergji Zaimi
37.4k●2●72●149

Interesting question that seems hard. The union of curves can't be dense in $\mathbb{R}^2$, because no Tutte polynomial of any graph (or for that matter, matroid) can have a root in the open first quadrant. But that argument doesn't work for $\mathbb{C}$. Do you know if $(2,2)$ is in the closure of the union of Tutte curves? If so, it would be interesting to see how. I'd also be curious about what happens if you replace "graph" with "matroid". – Jeremy Martin Jul 31 at 4:43

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