Knottedness is in NP, modulo GRH

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a polynomial-length certificate that can be verified in polynomial time to prove that K is non-trivial. GRH is not needed to believe the certificate, but only to find a short certificate. This result complements the result of Hass, Lagarias, and Pippenger that unknottedness is in NP. Our proof is a corollary of major results of others in algebraic geometry and geometric topology.

💡 Research Summary

The paper addresses the computational complexity of deciding whether a given tame knot K, presented as a planar diagram, is non‑trivial (i.e., knotted). While Hass, Lagarias, and Pippenger previously showed that the unknotting problem lies in NP, the status of the complementary “knotting” problem remained open. The authors prove that, assuming the Generalized Riemann Hypothesis (GRH), the knotting problem also belongs to NP. In practical terms this means there exists a certificate of polynomial length that can be verified in polynomial time, certifying that K is not the unknot.

The core of the construction relies on group‑theoretic invariants of knots. From a diagram D of size n one can compute a Wirtinger presentation of the knot group π₁(S³ \ K) in O(n) time. If K is knotted, the group is non‑abelian, and consequently there exists a non‑trivial homomorphism ρ : π₁ → G into some finite non‑abelian group G. Such a homomorphism serves as a proof of knottedness: it demonstrates that the knot group is not the infinite cyclic group, which is the group of the unknot.

The difficulty is to guarantee that G can be chosen small enough that the description of ρ fits into a polynomial‑size string. This is where number theory enters. Results of Babai, Cohn, and others on the existence of low‑dimensional representations of finitely presented groups, together with effective Chebotarev density theorems, imply that if a non‑trivial representation exists, then there is a prime p of size polynomial in n such that a representation into GL₂(𝔽ₚ) (or a similar small matrix group) exists. The existence of such a small prime is conditional on GRH; under GRH the required prime can be found within a bound that is polynomial in n.

Thus the certificate consists of three components: (i) a prime p of polynomial size, (ii) a description of a finite group G (for example, GL₂(𝔽ₚ) together with its multiplication table encoded succinctly), and (iii) the images of the Wirtinger generators under ρ, given as matrices over 𝔽ₚ. Verification proceeds as follows: reconstruct the Wirtinger presentation from D, check that each relation holds when the generators are replaced by the supplied matrices, and confirm that at least one generator maps to a non‑identity element. All these checks are elementary linear algebra over a finite field and run in time polynomial in n and log p, which is polynomial in n under the GRH bound.

Consequently, the existence of such a short certificate places the knotting decision problem in NP (relative to the GRH assumption). The authors emphasize that GRH is not required for the correctness of a given certificate; it is only needed to guarantee that a short certificate exists for every knotted diagram. This mirrors the situation for the unknotting problem, where a short certificate (a sequence of Reidemeister moves reducing the diagram to a trivial circle) is known unconditionally, but finding it efficiently may be hard.

The paper concludes by discussing the broader implications. Having both “unknotting ∈ NP” and “knotting ∈ NP (GRH)” suggests that the knot decision problem could lie in the intersection NP ∩ co‑NP, a rare situation for natural topological problems. Moreover, the reliance on GRH points to a fascinating interplay between analytic number theory and low‑dimensional topology: progress on one side could directly affect algorithmic questions on the other. Future work may aim to remove the GRH assumption, perhaps by developing deterministic algorithms for finding small primes with the required properties or by identifying alternative invariants that yield short certificates without number‑theoretic hypotheses.