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 (crossing switch) 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. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots: * Trefoil knot unknotting number 1 * Figure-eight knot unknotting number 1 * Cinquefoil knot unknotting number 2 * Three-twist knot unknotting number 1 * Stevedore knot unknotting number 1 * 6₂ knot unknotting number 1 * 6₃ knot unknotting number 1 * 7₁ knot unknotting number 3 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. (The unknotting number of the 1011 prime knot is unknown.) (Wikipedia).
