handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality obviously holds, so in particular these invariants place upper bounds on the genus. The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston. It has been shown that there are Seifert surfaces of the same genus that do not become isotopic either topologically or smoothly in the 4-ball. See also Crosscap number Arf invariant of a knot Slice genus References External links The SeifertView programme of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm. Geometric topology Knot theory Surfaces