Simon’s Epimorphism Conjecture for Knot Groups

Definitions and mathematical statement

For a classical knot $K\subset S^3$, let $\Gamma_K=\pi_1(S^3\setminus K)$, equivalently the fundamental group of its compact exterior. A nontrivial knot means one not isotopic to the unknot. Let \[\mathcal Q(K)=\{[\Gamma_J]:J\subset S^3\text{ is a knot and there exists a surjective homomorphism }\Gamma_K\twoheadrightarrow\Gamma_J\},\] where brackets denote abstract group-isomorphism class. Simon's conjecture asks whether \[\forall\text{ nontrivial knots }K\subset S^3:\qquad |\mathcal Q(K)|<\infty.\] The target knots are not required to be nontrivial by the supplied statement, and inclusion of the unknot group does not change the finiteness target. The assertion counts isomorphism classes of target knot groups, not the number of epimorphisms to a fixed target, and it requests no bound uniform over $K$. No hyperbolicity, primeness, peripheral-preservation, or nondegeneracy restriction is inserted into the general question. The supplied 2026-08-21 triage labels the record partially solved, reports the non-degenerate epimorphism case established, and says the general degenerate case remains open. It does not define non-degenerate, so the precise proved subclass is not reconstructed or imposed on the conjecture.

Short English statement

Must each nontrivial knot group have only finitely many knot-group quotients up to isomorphism? The question concerns target groups rather than the number of surjective maps, and retains arbitrary epimorphisms beyond the reported non-degenerate cases.

Sources

• Local UnsolvedMath registry, permanent ID 30001045, source problem catalogue entry, Simon's Epimorphism Conjecture for Knot Groups, proposed year 2008. The original attributes the conjecture to Simon; no separate individual proposer metadata is supplied.
• Supplied citation: Topologie (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/43, pp. 2419--2476: https://doi.org/10.4171/owr/2008/43. Supplied literature label: Oberwolfach report.
• Supplied literature label: Knot group epimorphisms II, https://arxiv.org/abs/0806.3223.
• Literature assessment date: 2026-08-21. The status and progress claims are supplied dated curation, not independently verified here.