Prime knots are determined by finite quotients

For a knot \(K \subset S^3\), let \(X_K\) be its exterior, \(G_K = \pi_1 X_K\) its group and \(\widehat{G}_K\) the profinite completion. Knot groups are finitely generated and residually finite, so two knot groups have the same finite quotients if and only if their profinite completions are isomorphic (Dixon, Formanek, Poland and Ribes).

Boileau and Friedl asked whether two prime knots whose groups have isomorphic profinite completions must be equivalent, allowing mirror images [BF, Question 1.3]. The following theorem answers this, and needs only one of the two knots to be prime.

Theorem. Let \(K \subset S^3\) be a prime knot and \(J \subset S^3\) a knot. If \(G_J\) and \(G_K\) have the same finite quotients, then \(J\) is equivalent to \(K\) or to its mirror image.

Both hypotheses are necessary. A knot and its mirror image have isomorphic groups, and so do the square and granny knots.

Background

The theorem was known for torus knots and the figure-eight knot (Boileau and Friedl; Bridson and Reid for the figure-eight among all 3-manifold groups), prime graph knots (Wilkes), two-bridge knots and Montesinos knots with three tangles (Cheetham-West and Xu), Montesinos knots with at least four tangles (Walton), and, by Xu’s theorem that lattices in \(\mathrm{PSL}_2(\mathbb{C})\) are profinitely rigid among lattices, all hyperbolic knots. For hyperbolic knots and hyperbolic-type satellites, whose boundary torus lies in a hyperbolic piece, Xu also showed that the completion detects that the 3-manifold is such a knot exterior, and matches up Dehn surgeries [X3, Theorem C].

The remaining knots are satellites whose exteriors contain a hyperbolic JSJ piece. For these we use the recognition theorem of Biswas and Singh [BS, Theorem A]. For oriented knots \(J, K\), the following are equivalent: \(J\) and \(K\) are equivalent as oriented knots; some isomorphism \(\widehat{G}_J \to \widehat{G}_K\) sends \(\mu_J\lambda_J\) to a conjugate of \(\mu_K\lambda_K\); some isomorphism sends \(\mu_J\) and \(\lambda_J\) to \(g\mu_Kg^{-1}\) and \(g\lambda_Kg^{-1}\) for a single \(g\). We use the last condition. Their theorem marks the completion; what follows shows that for prime knots the marking comes for free, up to reversal and mirroring. The proof below follows Xu’s proof that the A-polynomial of a prime knot is a profinite invariant up to mirror image, with the theorem of Biswas and Singh in place of the A-polynomial comparison.

Reductions

Trivial knots. If \(K\) is trivial, \(\widehat{G}_K \cong \widehat{\mathbb{Z}}\) is abelian, so \(G_J\), which embeds in \(\widehat{G}_J\), is abelian and \(J\) is trivial. If \(K\) is non-trivial, \(\widehat{G}_K\) contains the closure \(\widehat{\mathbb{Z}}^2\) of the peripheral subgroup but has abelianisation \(\widehat{\mathbb{Z}}\), so it is non-abelian. From now on both knots are non-trivial, and \(\Phi\colon \widehat{G}_J \to \widehat{G}_K\) is an isomorphism.

JSJ decomposition. By Wilton–Zalesskii and Xu [X1, Proposition 5.2, Theorem 5.5; X2, Lemma 9.7(2)], \(\Phi\) induces an isomorphism \(\theta\) of JSJ trees preserving the hyperbolic or Seifert type of each vertex, and isomorphisms \(\Phi_v\) of completed vertex groups. Each \(\Phi_v\) sends the closure of each JSJ-torus subgroup to a conjugate of the closure of the corresponding one. In particular, \(X_J\) is a graph manifold, hyperbolic, or mixed exactly when \(X_K\) is.

Graph manifolds. Wilkes proved [W2, Theorem 4.1] that if \(K\) is a prime graph knot and \(X_J\) a knot exterior with \(\widehat{G}_J \cong \widehat{G}_K\), then \(X_J \cong X_K\). By Gordon–Luecke, \(J\) is \(K\) or its mirror.

Hyperbolic exteriors. By Xu’s lattice theorem [X4, Theorem 1.2], \(G_J \cong G_K\), so \(X_J \cong X_K\) by Mostow–Prasad rigidity, and Gordon–Luecke applies.

It remains to treat mixed exteriors, which contain a hyperbolic piece and are not hyperbolic. Let \(P_K = \langle \mu_K, \lambda_K \rangle \cong \mathbb{Z}^2\) be the peripheral subgroup and \(\overline{P_K} \cong \widehat{\mathbb{Z}}^2\) its closure. The root is the vertex whose piece contains \(\partial X_K\).

Step 1: \(\theta\) maps the root to the root

The \(\Phi_v\) are only known to respect boundary tori coming from JSJ tori; profinite isomorphisms of Seifert pieces need not respect the others. Instead, let \(\partial^{\mathcal{T}} X_v\) be the union of the JSJ boundary tori of the piece \(X_v\). Then \(H_3(X_v, \partial^{\mathcal{T}} X_v; \mathbb{Z})\) is \(\mathbb{Z}\) if \(v\) is not the root and \(0\) if it is. Xu shows this group is determined by the profinite pair consisting of \(\widehat{G}_v\) and the closures of the JSJ-torus subgroups, which \(\Phi_v\) preserves. Hence \(\theta\) maps root to root [X2, Lemma 9.7(1)].

Step 2: an isomorphism respecting the boundary

Hyperbolic root. An isomorphism between completions of finite-volume hyperbolic 3-manifold groups induces a bijection of cusps, with conjugate cusp-subgroup closures [X1, Lemma 6.3]. \(\Phi_{\mathrm{root}}\) sends JSJ cusps to the corresponding JSJ cusps, and distinct cusps have non-conjugate closures [X1, Proposition 6.7(1)], so it sends \(\partial X_J\) to \(\partial X_K\). The other pieces have only JSJ boundary, so \(\Phi\) respects the peripheral structure [X1, Corollary 6.11]. Hence, after an inner automorphism, \(\Phi(\overline{P_J}) = \overline{P_K}\). Set \(J^0 = J\), \(K^0 = K\) and \(\Phi^0 = \Phi\).

Seifert root. By Budney [Bu, Theorem 4.18], the root is a torus knot exterior, a cable space or a composing space. The first is excluded because \(X_K\) is not Seifert fibred, the last because \(K\) is prime. So \(K\) is the \((p', q')\)-cable of a companion \(K^0\), oriented so that \(q' > 0\). The root has degree \(1\), a composing space has degree at least \(2\), and \(\theta\) preserves degree, so \(J\) is likewise the \((p, q)\)-cable of a companion \(J^0\) with \(q > 0\). Removing the roots, the JSJ data of \(\Phi\) induce an isomorphism \(\Phi^0\colon \widehat{G}_{J^0} \to \widehat{G}_{K^0}\) [X1, Proposition 3.11]. Every boundary torus of every piece of \(X_{J^0}\), including \(\partial X_{J^0}\), is a JSJ torus of \(X_J\), so \(\Phi^0\) respects the peripheral structure [X1, Corollary 6.11]. The companions need not be prime.

In both cases \(X_{J^0}\) is hyperbolic or mixed, and after an inner automorphism \(\Phi^0(\overline{P_{J^0}}) = \overline{P_{K^0}}\). Let \(A\) be the restriction. The closure of a boundary subgroup of a knot exterior or JSJ piece is its own normaliser and is abelian [X1, Proposition 6.7(2)], so \(A\) is independent of the inner automorphism. This is used again at the end.

Step 3: the boundary map is integral

By Xu’s peripheral regularity theorems [X2, Theorems 1.4 and 9.2], \(A\) is the completion of an isomorphism \(\psi\colon P_{J^0} \to P_{K^0}\). For hyperbolic \(X_{J^0}\), every profinite isomorphism is integral on each cusp. For mixed \(X_{J^0}\), this needs \(\Phi^0\) to respect the peripheral structure, and each Seifert piece to contain at most one boundary component.

In the mixed case, choose on each boundary torus of a Seifert piece a basis \((h, e)\) with \(h\) a regular fibre. There \(\Phi_v\) has matrix

\[\begin{pmatrix} \kappa_v & \rho_{v,T} \\ 0 & \tau_v \end{pmatrix}, \qquad \kappa_v, \tau_v \in \widehat{\mathbb{Z}}^\times,\ \rho_{v,T} \in \widehat{\mathbb{Z}},\]

with \(\kappa_v, \tau_v\) independent of \(T\) up to sign, and \(\kappa_w = \pm\tau_v\), \(\tau_w = \pm\kappa_v\) for adjacent Seifert vertices [X1, Lemma 9.3]. Each maximal Seifert subtree is adjacent to a hyperbolic vertex, where the map is integral, and an invertible upper-triangular integer matrix has diagonal \(\pm 1\). So \(\kappa_v, \tau_v = \pm 1\) throughout. Xu then shows the off-diagonal entry on the outer torus is an integer, using the relation \(\sum_T \rho_{v,T} = 0\) [X2, Lemma 9.10; X1, Lemma 8.15].

Step 4: meridian and longitude are preserved up to sign

This is [X2, Lemma 10.4]. The longitude generates the kernel of \(P \to H_1(X) \cong \mathbb{Z}\), which \(\Phi^0\) preserves in the completion. So \(\psi(\lambda_{J^0}) \in \widehat{\mathbb{Z}}\lambda_{K^0} \cap P_{K^0} = \mathbb{Z}\lambda_{K^0}\), and being primitive it is \(\lambda_{K^0}^{\pm 1}\).

For the meridian, let \(c = \psi^{-1}(\mu_{K^0})\). By right exactness of completion,

\[\widehat{\pi_1 X_{J^0}(c)} \;\cong\; \widehat{G}_{K^0} \big/ \overline{\langle\!\langle \mu_{K^0} \rangle\!\rangle} \;\cong\; \widehat{\pi_1 S^3} = 1,\]

where \(X_{J^0}(c)\) is the Dehn filling along \(c\). Its fundamental group is residually finite, hence trivial, so \(X_{J^0}(c) \cong S^3\) by the Poincaré conjecture, and \(c = \mu_{J^0}^{\pm 1}\) by Gordon–Luecke.

Conclusion

After an inner automorphism, \(\Phi^0(\mu_{J^0}) = \mu_{K^0}^{\varepsilon}\) and \(\Phi^0(\lambda_{J^0}) = \lambda_{K^0}^{\eta}\) with \(\varepsilon, \eta = \pm 1\). Reversing \(K\) inverts both, and mirroring \(K\) inverts the longitude. These operations act on \(K^0\) in the same way and preserve \(q' > 0\). After replacing \(K\) by its reverse, mirror or reversed mirror, \(\varepsilon = \eta = 1\), and [BS, Theorem A], which holds for all knots, gives that \(J^0\) and \(K^0\) are equivalent as oriented knots. If the root is hyperbolic, this completes the proof.

If the root is a cable space, let \(T\) be the JSJ torus between the root and the companion. The regular fibre \(h_J\) lies in \(\pi_1 T\) and equals \(\pm(p\mu_{J^0} + q\lambda_{J^0})\) there, and similarly for \(K\). Following the proof of [X2, Theorem 10.13], compare the two sides of \(T\) through the edge isomorphism \(\Phi_e\colon \overline{\pi_1 T_J} \to \overline{\pi_1 T_K}\). The congruence relations say that \(\Phi_{\mathrm{root}}\) and \(\Phi^0\) each restrict to \(\Phi_e\) on \(\overline{\pi_1 T}\), after conjugating by some element.

On the root side, the closure of the fibre subgroup is preserved by \(\Phi_{\mathrm{root}}\) [W1, Theorem 5.5] and is central, because the base orbifold of a cable space is orientable. So the conjugation fixes it, and \(\Phi_e(h_J) = \kappa h_K\) for some \(\kappa \in \widehat{\mathbb{Z}}^\times\).
On the companion side, the conjugating element carries \(\overline{P_{K^0}}\) to itself, so it lies in \(\overline{P_{K^0}}\) and acts trivially [X1, Proposition 6.7(2)]. So \(\Phi_e\) agrees with \(A\), which is now the identity in the marked bases.

Hence

\[\kappa\,(p', q') = \pm(p, q).\]

Since \(\widehat{\mathbb{Z}}\) is torsion-free, \(x \in \widehat{\mathbb{Z}}\) and \(cx \in \mathbb{Z}\) with \(c \neq 0\) imply \(x \in \mathbb{Z}\): reducing \(cx\) modulo \(c\) gives \(cx = ca\) with \(a \in \mathbb{Z}\). So \(\kappa q' = \pm q\) gives \(\kappa = \pm 1\), and \(q, q' > 0\) gives \((p', q') = (p, q)\). Thus \(J\) and \(K\) are the same cable of equivalent oriented knots. Undoing the reversal and mirroring, \(J\) is \(K\) or its mirror. ∎

Remarks

Primeness is used in Wilkes’ theorem and to exclude a composing root. It cannot be dropped in either case.
The hyperbolic and mixed cases rest on Xu’s lattice paper: directly through its Theorem 1.2, and through Biswas and Singh, who use its genuineness theorem [X4, Definition 6.1 and Proposition 16.3]. At the time of writing, both papers are preprints.

References

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