Homogeneous links and the Seifert matrix

TL;DR

This paper offers a geometric proof that the Seifert surface of homogeneous links has minimal genus, relating the Seifert matrix to the Seifert graph's block structure and classifying genus one homogeneous knots.

Contribution

It provides a geometric proof of minimal genus for homogeneous links and relates the Seifert matrix to the Seifert graph's block decomposition.

Findings

Seifert matrix can be arranged in a block triangular form

Diagonal blocks of the Seifert matrix have non-zero determinant

Complete classification of genus one homogeneous knots

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.