# Hyperbolic Dehn surgeries and SU(2)-representations

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 $$SU(2)$$?

• Just pointing out that the irreducible $SU(2)$ representations of the figure 8 knot complement were computed in this paper of Kirk and Klaasen, who showed that there is a circle's worth: mathscinet.ams.org/mathscinet-getitem?mr=1054574 I believe one could verify directly from this that all non-trivial Dehn fillings admit an irreducible rep. via an intersection argument, but I haven't attempted this myself. – Ian Agol May 3 '19 at 3:47

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. 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) 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.
• I'm claiming this fact for any rational homology sphere $Y$ with the additional hypothesis that $\pi_1(Y)$ is "cyclically finite", a condition which ensures that the Chern-Simons functional used to define $I^\#(Y)$ is Morse-Bott at the reducible representations. (See Theorem 4.6 of that paper for the precise statement.). When $Y$ is $p/q$-surgery on a knot $K$, Boyer--Nicas proved this hypothesis equivalent to the condition that no root of $\Delta_K(t^2)$ is a $p$th root of unity. – Steven Sivek Mar 7 '19 at 11:43