Kirby Problem 1.14

Definitions and mathematical statement

Let $K\subset S^3$ be a smooth nontrivial knot. For $r=p/q\in\mathbb Q$, with $p,q$ coprime and $q>0$, write $S^3_r(K)$ for Dehn surgery along $p\mu+q\lambda$, where $\mu$ is the meridian and $\lambda$ is the longitude having zero linking number with $K$. Define \[ \operatorname{SU}(2)=\{A\in M_2(\mathbb C):A^*A=I,\ \det A=1\}. \] A representation $\rho:\pi_1(S^3_r(K))\to\operatorname{SU}(2)$ has nonabelian image if some two elements of its image do not commute.

Determine exactly the set \[ \mathcal R=\{r\in\mathbb Q\setminus\{0\}:\text{ every nontrivial }K\subset S^3 \text{ admits such a nonabelian }\rho\}. \] The order of quantifiers matters: one slope must work for every nontrivial knot, while the representation can depend on both slope and knot. For a slope outside $\mathcal R$, a counterexample must have only abelian $\operatorname{SU}(2)$ images, not simply fail a particular construction of representations. Replacing a knot by its mirror reverses the surgery slope, so the classification is symmetric under $r\mapsto-r$.

Short English statement

Which nonzero rational surgeries on every nontrivial knot have a fundamental group with a nonabelian representation into $\operatorname{SU}(2)$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2673 (KP-1.14), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.14. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.