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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 $SU(2)$?

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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.

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  • $\begingroup$ You are saying that manifolds, which are not instanton L-spaces, must admit a nontrivial representation of their fundamental group to SU(2)? $\endgroup$ – ThiKu Mar 7 at 10:34
  • $\begingroup$ 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. $\endgroup$ – Steven Sivek Mar 7 at 11:43

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