Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi\_1(S^3-K)\to SL(2,{\mathcal O}\_3)\subset SL(2,C)$. This representation is not conjugate into $SU(2)$, because traces are not real. But, of course, one can get nontrivial $SU(2)$-representations factoring over the abelianization $Z$ of $\pi\_1(S^3-K)$.


Thurston goes on to prove that almost all Dehn surgeries at the figure eight knot complement admit a hyperbolic structure, hence a faithful, discrete representation $\pi\_1(S^3-K)\to PSL(2,C)$, which by Culler lifts to $SL(2,C)$ and again is not conjugate into $SU(2)$.


On the other hand, Kronheimer and Mrowka show that for Dehn surgery coefficients $\vert\frac{p}{q}\vert\le 2$, the Dehn surgered manifold admits a noncyclic representation to $SU(2)$. This is, in a sense, complementary to Thurston‘s result, who proved his result for sufficiently large Dehn surgery coefficients.


Question: is it known whether Dehn surgeries at the figure eight knot complement, with sufficiently large Dehn surgery coefficients, admit nontrivial (not necessarily faithful) representations to $SU(2)$?



Hyperbolic Dehn surgeries and SU(2)-representations
All Dehn surgeries on the figure eight knot $K$ admit irreducible $SU(2)$ representations. This can be proved using Corollary 4.8 of my paper with John Baldwin, "Stein fillings and $SU(2)$ representations", [arXiv:1611.05629](https://arxiv.org/abs/1611.05629). The case of 0-surgery follows from Kronheimer and Mrowka's work, as you mention, so I'll omit it.


Suppose for some $\frac{p}{q}$ that there are no irreducible representations $\pi\_1(S^3\_{p/q}(K)) \to SU(2)$; we can take $\frac{p}{q}>0$ without loss of generality since $K$ is amphichiral. Then the corollary says that either some root of $\Delta\_K(t^2) = -t^2+3-t^{-2}$ is a $p$th root of unity, which by inspection doesn't happen, or $S^3\_{p/q}(K)$ is an "instanton L-space", meaning that the framed instanton homology $I^\#(S^3\_{p/q}(K))$ has rank $p$.


If $S^3\_{p/q}(K)$ is an instanton L-space, then Theorem 4.20 of this paper asserts that $I^\#(S^3\_n(K))$ likewise has rank $n$ for all sufficiently large integers $n$. At this point, an argument involving the surgery exact triangle and the adjunction inequality (see Proposition 9.4 of [arXiv:1710.01957](https://arxiv.org/abs/1710.01957)) says that since $K$ has slice genus 1, we can in fact take $n=1$. But $S^3\_1(K)$ is the Brieskorn sphere $\Sigma(2,3,7)$, whose framed instanton homology has rank 3, so none of the surgeries on $K$ are L-spaces after all.


