The unknotting number of a knot K is the smallest number of times u(K) the knot must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in general difficult to determine exact values. Many unknotting numbers can be determined from a knot's knot signature. A knot with unknotting number 1 is a prime knot. It is not always true that the unknotting number is achieved in a projection with the minimal number of crossings. The unknotting number is not additive for two combined knots. In particular, for the combined 7_1 torus knot and its mirror image, u(7_1 #(7_1)^_) = 5.

algebraic unknotting number | Bennequin's conjecture | knot signature | Milnor's conjecture | slice-Bennequin inequality