Hard unknot

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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]

Contents

Background

An easy unknot, reduced to a trivial diagram by a type I Reidemeister move. Easyunknot.png
An easy unknot, reduced to a trivial diagram by a type I Reidemeister move.

An unknot is a closed loop in three dimensions that does not contain knots and can, in principle, be stretched out into a circle without any part of the loop passing through another part. A diagram of an unknot is a projection of its three dimensional shape onto two dimensions, where the loop can appear to cross over itself. At each crossing where two parts of the curve intersect, the diagram will show which part of the curve passes over or under the other. To demonstrate whether any given diagram is an unknot, a sequence of Reidemeister moves must be applied to the diagram to eliminate all the crossings until the diagram is a circle, known as simplifying the diagram. This typically involve passing parts of the diagram over each other (Reidemeister types II and III), or untwisting loops (type I). While an individual diagram may be simplified in a small number of Reidemeister moves, it is very difficult to know how many moves this will take for an arbitrary diagram.

Examples

From top to bottom, the Goeritz, Culprit, and Monster unknots HardUnknots.png
From top to bottom, the Goeritz, Culprit, and Monster unknots

Early examples of hard unknot diagrams were created by Lebrecht Goeritz in 1934. A diagram known as the Goeritz unknot contains 11 crossings but requires an additional crossing to be created in order to simplify it. [2] Another early diagram is known as "the Culprit" and was created by Ken Millett in 1988. [3] It contains 10 crossings. At least two additional crossings must be introduced, making the diagram reach at least 12 crossings, before the knot can be untied using planar Reidemeister moves. (Note, however, only one new crossing needs to be introduced when working with spherical Reidemeister moves.) Many other examples exist such as "the Monster" created by Rob Scharein, who used a physics engine to show that hard unknots could be simplified. [4] A 2025 computational study found 2.6 million cases of hard unknot diagrams that could not be simplified by available algorithms, but were determined to be unknotted through the calculation of knot invariants. [5]

Unknotting on a Sphere

If a diagram lies on the surface of a sphere rather than a plane, unknotting can be simpler as part of the diagram may (for example) slide over the North Pole, pass over the equator, and be brought up from the South Pole. In the case of both the Goeritz unknot and the Culprit, only one extra crossing (rather than two) is required on a sphere, and the Monster no longer requires additional crossings. In 2021 it was demonstrated that no previously published example of a hard unknot requires more than one additional crossing on a sphere. [6] Computational methods were used to create new hard unknot diagrams that require at least three additional crossings, on either a sphere or a plane, currently the hardest known unknots.

References

  1. ↑ Henrich, Allison; Kauffman, Louis (2024). "Unknotting Unknots". American Mathematical Monthly: 379–390. Retrieved 2025-10-22.
  2. ↑ Goeritz, Lebrecht (1934). "Bemerkungen zur knotentheorie". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German). 10 (1): 201–210. doi:10.1007/BF02940674. ISSN   0025-5858 . Retrieved 2025-10-22.
  3. ↑ Kauffman, Louis H.; Lambropoulou, Sofia (2006-01-22). "Hard Unknots and Collapsing Tangles". arXiv.org. Retrieved 2025-10-22.
  4. ↑ Scharein, Robert Glenn. "Interactive topological drawing". University of British Columbia. University of British Columbia. doi:10.14288/1.0051670 . Retrieved 2025-10-22.
  5. ↑ Applebaum, Taylor; Blackwell, Sam; Davies, Alex; Edlich, Thomas; Juhász, András; Lackenby, Marc; Tomašev, Nenad; Zheng, Daniel (2025-08-18). "The Unknotting Number, Hard Unknot Diagrams, and Reinforcement Learning". Experimental Mathematics: 1–19. doi: 10.1080/10586458.2025.2542174 . ISSN   1058-6458 . Retrieved 2025-10-22.
  6. ↑ Burton, Benjamin A.; Chang, Hsien-Chih; Löffler, Maarten; Maria, Clément; de Mesmay, Arnaud; Schleimer, Saul; Sedgwick, Eric; Spreer, Jonathan (2024-07-02). "Hard Diagrams of the Unknot". Experimental Mathematics. 33 (3): 482–500. doi:10.1080/10586458.2022.2161676. ISSN   1058-6458 . Retrieved 2025-10-22.
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