Does the Jones polynomial detect the unknot?

The Jones polynomial, introduced by Vaughan Jones in 1985, transformed knot theory and tied it to quantum field theory. The unknot has Jones polynomial 1, and nobody knows whether some knotted loop shares that value; computer checks have ruled out every knot with up to 24 crossings. The task is to find such a knot, or to prove that none exists.

Problem

Let $V_K(t)$ be the Jones polynomial of a knot $K \subset S^3$, normalised so that $V_{\mathrm{unknot}}(t) = 1$.

Question (Jones unknot conjecture). If $V_K(t) = 1$, must $K$ be the unknot?

A counterexample is a knot $K$ that is genuinely knotted, for instance with Seifert genus $g(K) > 0$, and has $V_K(t) = 1$.

Background

Jones found his invariant through the theory of von Neumann algebras [1], and Witten soon reinterpreted it through Chern–Simons quantum field theory [2]. It distinguishes a great many knots, but whether it can recognise the simplest one is a long-standing question. A related invariant does: Khovanov homology, a categorification of the Jones polynomial, detects the unknot (Kronheimer and Mrowka, 2011) [3]. A counterexample to the Jones unknot conjecture would therefore have to fool the polynomial while being caught by its categorification.

What is known

• No nontrivial knot with at most 24 crossings has Jones polynomial 1 (Tuzun and Sikora, 2021) [4].
• Khovanov homology detects the unknot [3].
• Epoch notes that even if a counterexample exists, the smallest one might be too large to verify computationally.

What would count as progress

• A planar diagram (PD code) of a knot with Jones polynomial exactly 1, together with a certificate that it is knotted, for example positive Seifert genus read off from knot Floer homology. Both computations are routine for diagrams of moderate width.
• Epoch scores submissions with its own verifier: a reduced PD code (no simplifying Reidemeister I or II move) with at most 100 crossings. The verifier computes the Jones polynomial and knot Floer homology, allowing up to 30 minutes in total, so diagrams should be presented with small width, for instance as closed braids on few strands.
• A proof that the Jones polynomial detects the unknot would settle the question the other way.

References

1. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111. doi:10.1090/S0273-0979-1985-15304-2
2. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399. doi:10.1007/BF01217730
3. P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, Publ. Math. IHÉS 113 (2011), 97–208. doi:10.1007/s10240-010-0030-y
4. R. E. Tuzun and A. S. Sikora, Verification of the Jones unknot conjecture up to 24 crossings, J. Knot Theory Ramifications 30(3) (2021), 2150020. doi:10.1142/S0218216521500206

Source

Imported from FrontierMath: Open Problems, jones-unknot, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.