# Three-Genus

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 computation of the Ozsvath-Szabo
knot Floer homology. For twelve crossing knots, original data was provided by Stoimenow, available at
stoimenov.net/stoimeno/homepage/ptab/index.html.

## Specific Knots

11n_{34}, 11n_{42}, 11n_{45}, 11n_{67}, 11n_{73}, 11n_{97},
11n_{152}

not equal to half degree of Alexander Polynomial