Homogeneous links and the Seifert matrix

Abstract

Homogeneous links were introduced by Peter Cromwell, who pr oved that the projection surface of these links, that given by the Seifert al- gorithm, has minimal genus. Here we provide a different proof , with a geometric rather than combinatorial flavor. To do this, we fir st show a direct relation between the Seifert matrix and the decompo sition into blocks of the Seifert graph. Precisely, we prove that the Sei fert matrix can be arranged in a block triangular form, with small boxes in th e diagonal corresponding to the blocks of the Seifert graph. Then we pro ve that the boxes in the diagonal has non-zero determinant, by looking a t an explicit matrix of degrees given by the planar structure of the Seifer t graph. The paper contains also a complete classification of the homogen eous knots of genus one.