Unknotting Number = 1

Devise an algorithm that decides whether a knot has unknotting number equal to 1.

Notability: Breakthrough

Solved: No

Contributor: Joel Hass

Field: Topology / Geometry

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About the problem Attempts by AI AI prompts Mathematician survey

About the problem

The unknotting number of a knot is a classic invariant. It is defined as the smallest number of crossing changes required to change a diagram of the knot into a diagram of the unknot. This problem is to devise an algorithm that decides whether a diagram depicts a knot with unknotting number one.

This problem is one of the fundamental questions in low-dimensional topology and a solution would be a major result. The problem sits at an interesting point in terms of complexity: there are polynomial time algorithms for determining if the unknotting number of a knot is zero, but the general problem of determining the unknotting number of a knot is NP-hard — and is not even known to be decidable. The problem author is optimistic that the case of unknotting number equal to one is at least decidable, and even computationally manageable for diagrams of moderate size.

Rather than attempt full theoretical verification, we test proposed algorithms on a hidden challenge set of knots with known unknotting numbers. While we hope that perfect performance on this challenge set would be indicative of a conceptual breakthrough, it is possible that an AI system will throw together many ad hoc approaches and succeed. If so, we may repair the problem by generating still more challenging examples.

Warm-up: we ask for the case of unknotting number = 0 (detecting unknots), which is doable with existing techniques and software, though it can be computationally expensive in some cases.

Full write-up

Attempts by AI

We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.

Model

Problem

Solved

Logs

GPT-5.2 Pro

Warm-up

Yes

GPT-5.2 Pro

Full Problem

Gemini 3 Deep Think

Full Problem

AI Prompts

Warm-up

Produce an algorithm that takes as input a diagram of a knot and returns True if the unknotting number of the knot represented by that diagram is 0 and False otherwise. For a knot with up to 100 crossings, the algorithm must complete in under an hour when run on a typical laptop.

Provide your solution as a self-contained Python script. Accept 1-indexed planar crossing diagrams as the input format.

Example input: [[4,2,5,1],[10,6,11,5],[8,3,9,4],[2,9,3,10],[11,16,12,17],[7,15,8,14],[15,7,16,6],[13,20,14,21],[17,22,18,1],[21,18,22,19],[19,12,20,13]]

Full problem

Produce an algorithm that takes as input a diagram of a knot and returns True if the unknotting number of the knot represented by that diagram is 1 and False otherwise. For a knot with up to 100 crossings, the algorithm must complete in under an hour when run on a typical laptop.

Provide your solution as a self-contained Python script. Accept 1-indexed planar crossing diagrams as the input format.

Example input: [[4,2,5,1],[10,6,11,5],[8,3,9,4],[2,9,3,10],[11,16,12,17],[7,15,8,14],[15,7,16,6],[13,20,14,21],[17,22,18,1],[21,18,22,19],[19,12,20,13]]

Mathematician survey

The author assessed the problem as follows.

# of mathematicians highly familiar with the problem: a majority of those working in a subfield (≈100)

# of mathematicians who have made a serious attempt to solve the problem: 10–50

Rough guess of how long it would take an expert human to solve the problem: 3–10 years

Notability of a solution: a major advance in a broader field; one of the best results of the year

A solution would be published: in a leading general journal

Likelihood of a solution generating more interesting math: very likely: any solution would lead to a rich set of follow-on work

Probability that the problem is solvable as stated: 80–95%