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The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity element with respect to the knot sum operation.

Unknotting problem
Main article: Unknotting problem

Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Currently there are several well-known unknot recognition algorithms (not using invariants), but they are either known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as finite type invariants, are a complete invariant of the unknot, but knot Floer homology is known to detect the unknot. Even if they were, the problem of computing them efficiently remains.

Examples

Many useful practical knots are actually the unknot, including all knots which can be tied in the bight. Other noteworthy unknots are those that consist of rigid line segments connected by universal joints at their endpoints (linkages), that yet cannot be reconfigured into a convex polygon, thus acquiring the name stuck unknots.

Invariants

The Alexander polynomial, Conway polynomial and Jones polynomial of the unknot are trivial:

$$\Delta(t) = 1,\quad \nabla(z) = 1,\quad V(q) = 1.$$

No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the Kinoshita-Terasaka knot and Conway knot (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.

The knot group of the unknot is an infinite cyclic group, and the knot complement is homeomorphic to a solid torus.

Knot (mathematics)
Unknotting problem