Consider a Dehn filling of the figure eight knot with coprime parameters (p,q) denoted by M. I am interested in the representation space of M in $SU_2$.

I wonder wether all such representations are infinitesimally isolated or said differently $H^1(M,Ad_\rho)=0$ for all non central $\rho:\pi_1(M)\to SU_2$.

The representation space of the figure eight knot is described (except for the 2 isolated solutions) by the curve $$F(x,y)=cos(x)+1-cos(4y)+cos(2y)=0$$ where $x$ and $y$ are the angles of the meridian and the longitude respectively.

The question is equivalent to saying that the line of equation $px+qy=0$ meets transversely the curve $F(x,y)=0$. This is clear for geometric reasons if $|p/q|<\sqrt{20}$ and can be proven for $|p/q|>\sqrt{20}$ by showing that the polynomial $$\Phi=X^{2p}-X^{p+4q}+X^{p+2q}+2X^p+X^{p-2q}-X^{p-4q}+1$$ has simple roots except $-1$ if $p$ is odd and $\pm i$ if $4|p$.

This is a simple test which I ran with my computer for all $p<200$: the answer is yes. But I don't find a general argument. Any help is welcome!

Julien Marché

ps: of course a theoretical argument would be better but I don't know any.