The 3-genus of a knot is defined to be the minimal genus of a Seifert
surface for a knot. The three genus is bounded below by half the
degree of the Alexander polynomial.
For prime knots of 10 or fewer crossings, this bound is always
realized by a surface. For knots of 11 crossings there are seven
counterexamples: 11n_{34} (g=3), 11n_{42} (g=2), 11n_{45} (g=3),
11n_{67} (g=2), 11n_{73} (g=3), 11n_{97} (g=2) and 11n_{152} (g=3).

The evaluation of the genus was done by Jake Rasmussen, using a computer assisted computations of the Ozsvath-Szabo knot Floer homology. For twelve crossing knots, original data was provided by Stoimenow, available at www.ms.u-tokyo.ac.jp/~stoimeno/ptab/index.html.

Further information on particular knots.