Manifolds and cell complexes Knot theory Example

Unknot

Example

The unknot is the round circle $U = \set{(x, y, 0) \in \R^3 : x^2 + y^2 = 1},$ the image of the map $t \mapsto (\cos t, \sin t, 0)$ , which is continuous and injective on the circle $\R / 2\pi\Z$ . So $U$ is a knot, and it is tame since it is equivalent to a square, a closed polygon. Any knot equivalent to $U$ is also called an unknot, or a trivial knot.

Remarks

The unknot bounds the flat unit disk, a Seifert surface of genus $0$ , and a knot is trivial exactly when it bounds an embedded disk. Its knot group is $\Z$ , its crossing number is $0$ , and its Alexander and Jones polynomials are $1$ . A tame knot with knot group $\Z$ is the unknot. Recognizing the unknot from a diagram is the unknotting problem; Wolfgang Haken showed that it is decidable. Some diagrams of the unknot are complicated, and untangling them may first require more crossings.

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Tag	`1TQ06R4Q66`
URI	`m2://1TQ06R4Q66`
Kind	example
Writing	W0 stub
MSC	57K10
Instance of	Knot
Same as	Wikidata Q1188344 Wikipedia Unknot
Changed	2 November 2026

Unknot

Example

Remarks

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