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: 11n34 (g=3), 11n42 (g=2), 11n45 (g=3), 11n67 (g=2), 11n73 (g=3), 11n97 (g=2) and 11n152 (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

11n34, 11n42, 11n45, 11n67, 11n73, 11n97, 11n152
not equal to half degree of Alexander Polynomial