unknotting

Unknotting is a concept in knot theory describing the transformation of a knot into the unknot, the simplest knot, through local changes and ambient isotopy. In practice, the most common local operation is a crossing change, which switches an over-crossing to under-crossing or vice versa in a knot diagram. Along with Reidemeister moves, which rearrange a diagram without changing its topological type, a knot can be manipulated toward the unknot in a finite sequence.

The unknotting number u(K) of a knot K is the minimum number of crossing changes required to convert K into the unknot. This number is an invariant of the knot, meaning it does not depend on the chosen diagram. Some well-known knots have low unknotting numbers: the trefoil and the figure-eight knot each have u(K) = 1. Other knots require more crossing changes, and many knots with the same crossing number can have different unknotting numbers.

Determining u(K) is generally challenging. Invariants such as the knot determinant and the signature can provide lower bounds in some cases, but no simple formula exists for all knots. The study of unknotting intersects several areas of knot theory, including knot concordance and 3-manifold topology; it also informs practical operations such as simplifying knot diagrams or understanding the complexity of a knot.

See also: Reidemeister moves, crossing change, unknot, knot invariant, knot theory.