The genus of a knot surface is much more complicated than the other knot properties discussed so far.
First, the oriented knot diagram must be broken down into Seifert circles. To obtain the seifert circles, each crossing must be removed. To do this, each crossing is converted to an L0 type crossing as in the Jones polynomial. These circles are allowed to be nested. Then, substitute into the equation below the number of Seifert circles, s, and the number of crossings, d.
g(K) = (d - s + 1) ÷ 2
g(K) is the genus of the given projection. Now, the genus of a knot is the smallest possible value that this equation can return, so that g(K) is actually less than or equal to (d - s + 1) ÷ 2. Unfortunately, the only general method of determining the minimal genus, designed by W. Hacken in 1961, is impractical.