Talk:Hard unknot
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 | The content of the Hard unknot page was merged into Unknot#Hard unknot on 14 March 2026. For the contribution history and old versions of the merged article, please see its history. |
Lead
[edit]Thank you for making this page!
The lead currently reads:
> In the mathematical field of knot theory, a hard unknot is a diagram of the unknot for which proving that it is unknotted is difficult. Hard unknot diagrams typically have at least ten crossings, and the difficulty arises both from the human perception of knottedness as well as from the number of Reidemeister moves required to reduce the diagram to that of a circle. Typically, a hard unknot diagram requires additional crossings to be introduced before the number of crossings can be reduced to zero. These diagrams are of importance to the field of knot theory because they can serve as cases for which conjectures about unknotting algorithms can be tested.[1]
I disagree with all of this. A hard unknot is usually taken to mean an unknot for which untying it using the Reidemeister moves requires first introducing more crossings before the number can be decreased to 0 eventually. Being a hard unknot is not related to human perception of difficulty. It is also not clear to me that the main reason people care about hard unknots is because of unknotting algorithms.
I think this would be better:
> In the mathematical field of knot theory, a hard unknot is a diagram of an unknot for which additional crossings must be introduced when untying the knot (using Redemeister moves) before the number of crossings can be reduced to zero. Mathwriter2718 (talk) 14:00, 26 October 2025 (UTC)
- I am not married to the current wording. There are some statements in the references about intuition (e.g. the Kauffman/Lambropoulou one states "Can you tell whether this knot is knotted or not? It requires quite some intuition for topology to just look at a knot and know if it is really knotted." but you're right that the strict definition is about requiring more crossings. The page also needs a reason for why hard unknots are notable, Kauffman/Lambropoulou state that introducing hard unknots to people demonstrates the need for a theory of knots. So if statements are being removed, they should be replaced with some other justification. ProfKnots (talk) 14:41, 27 October 2025 (UTC)
Spherical vs planar moves
[edit]The section on unknotting on the sphere should be merged into the section on background, and the discussion on what specific unknots are hard on the sphere should be discussed as those specific examples come up instead of in a separate section. Mathwriter2718 (talk) 14:20, 26 October 2025 (UTC)
Feedback from New Page Review process
[edit]I left the following feedback for the creator/future reviewers while reviewing this article: Thanks for helpfully creating this page. It's so well-written, that I think that its better on unknot, where is provides helpful references and background, as well as further details!