Unknotting number
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number, then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936.
Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The unknotting number is not additive under connected sum, although that possibility, implicit in and explicitly asked by Gordon in 1977 and many others, was not resolved until 2025. A counterexample showed that the unknotting number of the connected sum of 71 and its mirror image was one less than the sum of the numbers from its components.
The following table show the unknotting numbers for the first few knots:
In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:
- The unknotting number of a nontrivial twist knot is always equal to one.
- The unknotting number of a -torus knot is equal to.
- The unknotting numbers of prime knots with nine or fewer crossings have all been determined.