Homogeneous links and the Seifert matrix

Abstract: Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix and the decomposition into blocks of the Seifert graph. Precisely, we prove that the Seifert matrix can be arranged in a block triangular form, with small boxes in the diagonal corresponding to the blocks of the Seifert graph. Then we prove that the boxes in the diagonal has non-zero determinant, by looking at an explicit matrix of degrees given by the planar structure of the Seifert graph. The paper contains also a complete classification of the homogeneous knots of genus one.