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.

The Rasmussen Invariant, Four-genus, and Three-genus of an Almost Positive Knot Are Equal

“…The proofs of Theorem 1.1 and Corollaries 1.2 and 1.3 are given in Section 3. Homogeneous links are introduced by Cromwell [7] (see also [2,4,8]). From its definition, any positive knot is homogeneous.…”

On a conjecture by Kauffman on alternative and pseudoalternating links

“…[6] A genus one knot is homogeneous if and only if it belongs to one of the two following classes of knots:1.-Pretzel knots with diagram P (a, b, c), where a, b, c are odd integers with the same sign.2.-Pretzel knots with diagram P (m, e, k . .…”