Knottedness is in NP, modulo GRH

📝 Abstract

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.

💡 Analysis

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.

📄 Content

Knottedness is in NP, modulo GRH Greg Kuperberg∗ Department of Mathematics, University of California, Davis, CA 95616 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. 1. INTRODUCTION The algorithmic complexity of unknottedness is a long- standing open problem. In other words, given a knot K de- scribed by a knot diagram or by a triangulation of its com- plement, is there a fast algorithm to decide whether K is the unknot? (The question makes sense for tame knots; all knots in this article will be tame.) Haken [15] was the first to show that there is any algorithm. Currently it is not known whether unknottedness can be decided in polynomial time. The ques- tion can be written Unknottedness ?∈P, since P is the class of yes-no functions (or yes-no questions or decision problems) on input strings that can be computed in polynomial time. Welsh [32] proposed the study of qualitative rather than quantitative bounds on the algorithmic complexity of prob- lems in knot theory (and by extension, in low-dimensional topology). In response, Hass, Lagarias, and Pippenger [18] showed that unknottedness is in the complexity class NP. This is the class of yes-no questions for which an answer of yes can be confirmed in polynomial time with the aid of an auxiliary string called a witness, a proof, or a certificate. For exam- ple, the question of whether an integer N (written in binary) is composite is trivially in NP, because a certificate can consist of a factorization N = AB that proves that N is composite. By definition, the class coNP is the class of questions whose negations, with no and yes switched, are in NP. In other words, a problem is in coNP if there is a certificate for no rather than yes. For example, it is a non-trivial result that pri- mality, the negation of compositeness, is in NP, equivalently that compositeness is in coNP [29]. This result began an en- couraging chain of results. The computational complexity of primality was improved qualitatively in stages until finally it was established that primality is in P [2, 4, 13, 26, 30]. How- ever, there are other problems that are in both NP and coNP that are thought to be hard, for instance, determining whether an integer is the product of two primes. Theorem 1.1. Let K ⊂S3 be a knot described by a knot dia- gram, a generalized triangulation, or an incomplete Heegaard ∗greg@math.ucdavis.edu; Partly supported by NSF grant DMS CCF- 1013079 diagram. Then the assertion that K is knotted is in NP, assum- ing the generalized Riemann hypothesis (GRH). Together with Hass-Lagarias-Pippenger, we can restate the result as Unknottedness ∈NP∩coNP, assuming GRH. The complexity theory significance of this result is that unknottedness is not NP-hard, assuming stan- dard conjectures in both number theory and complexity the- ory. (See Section 2.) It is instead in the class of intermediate problems, such as graph isomorphism and factoring integers, that either have undiscovered polynomial-time algorithms or are hard for some other reason. We clarify the sense in which Theorem 1.1 depends on the generalized Riemann hypothesis. Since the role of the verifier for a problem in NP is to evaluate a proof of “yes”, the veri- fier might need to assume a conjecture such as GRH to believe the proof. This is not the case for our construction in Theo- rem 1.1. Our certificates are unconditionally convincing, and they always exist. The only role of GRH is to establish that the certificate has polynomial length. Instead of assuming all of GRH, we can assume a much weaker corollary, Theorem 3.2. The corollary asserts that for every non-constant univariate in- teger polynomial h(x), there is a moderately small prime p such that h(x) has a root in Z/p. Our proof of Theorem 1.1 quickly follows from major re- sults of others. Kronheimer and Mrowka [21] showed that if K is a non-trivial knot, then there is a non-commutative repre- sentation of ρC : π1(S3 \K) →SU(2) ⊂SL(2,C). Then, simply because the equations for the representation are algebraic, the complex numbers can be replaced by a finite field Z/p. Koiran [20] showed that if a polynomial-length set of algebraic equations has a complex solution, and if GRH is true, then there is a suitable prime p with only polynomially many digits. Thus, the certificate is a prime p and a 2 × 2 matrix over Z/p for each generator of the knot group. The verifier must check that the generator matrices satisfy the re- lations of

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