unknot

In knot theory, the unknot is the simplest knot. Formally, it is an embedding f: S^1 → R^3 (or S^3) whose image can be deformed, via an ambient isotopy of R^3, to a standard round circle. Two knots are equivalent if there exists such an ambient isotopy taking one embedding to the other.

In knot diagrams, the unknot corresponds to a closed curve that can be transformed into a simple loop through Reidemeister moves, which are local diagrammatic moves that do not change the ambient isotopy class. A knot is identified as the unknot if its diagram can be reduced to a circle by a finite sequence of these moves.

Properties and invariants: the complement of the unknot is a solid torus, and the knot group of the unknot is isomorphic to the integers Z. Among common invariants, the Alexander polynomial evaluates to 1 for the unknot, and many quantum invariants take their simplest value on the unknot; however, these invariants do not constitute a complete test for unknottedness, since nontrivial knots can share some of these values.

Unknot recognition and algorithms: the central decision problem is to determine whether a given knot diagram represents the unknot. This problem is known to be decidable, with procedures based on 3-manifold topology and normal surface theory providing finite algorithms. In practice, results from diagram simplification and checks against several invariants are commonly used, especially for manual or computational knot classifications.

The unknot serves as a baseline object in knot theory and underpins many theoretical and computational results, including the study of knot complements and the limits of knot invariants in distinguishing knot types.