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

Specific Knots

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