nLab unknot

Idea

The trivial knot is called the unknot.

Similarly, a link whose components are unknots not linked with each other is called an unlink.

Definition

An unknot is any knot isotopic to the flat circle embedded in the $(x,y)$-plane of $\mathbb{R}^3$.

Related concepts

• trefoil knot

• figure eight knot

References

See also

• Wikipedia, Unknot

Computation of the perturbative Wilson loop observable (universal Vassiliev invariant) of the unknot (“Wheels theorem”):

• Dror Bar-Natan, Thang T Q Le, Dylan Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. Volume 7, Number 1 (2003), 1-31 (euclid.gt/1513883092)

following

• Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, Dylan Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot (q-alg/9703025)

On machine learning the detection of unknots:

• Sergei Gukov, James Halverson, Fabian Ruehle, Piotr Sułkowski, Learning to Unknot, Mach. Learn.: Sci. Technol. 2 (2022) 025035 [arXiv:2010.16263, doi:10.1088/2632-2153/abe91f]

Exposition in:

• Sergei Gukov, Machine learning and hard problems in topology, talk at QFT and Cobordism, CQTS (Mar 2023) [web]