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Homogeneous links were introduced by Peter Cromwell, who proved that theprojection surface of these links, that given by the Seifert algorithm, hasminimal genus. Here we provide a different proof, with a geometric rather thancombinatorial flavor. To do this, we first show a direct relation between theSeifert matrix and the decomposition into blocks of the Seifert graph.Precisely, we prove that the Seifert matrix can be arranged in a blocktriangular form, with small boxes in the diagonal corresponding to the blocksof the Seifert graph. Then we prove that the boxes in the diagonal has non-zerodeterminant, by looking at an explicit matrix of degrees given by the planarstructure of the Seifert graph. The paper contains also a completeclassification of the homogeneous knots of genus one.

Homogeneous links and the Seifert matrix