Homogeneous links and the Seifert matrix

Pedro M. González Manchón

Vol. 255 (2012), No. 2, 373–392

DOI: 10.2140/pjm.2012.255.373

Abstract

Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, 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 have nonzero determinant, by looking at an explicit matrix of degrees given by the planar structure of the Seifert graph. The paper also contains a complete classification of homogeneous knots of genus one.

Keywords

homogeneous link, projection surface, Seifert graph, Seifert matrix, Conway polynomial, knot genus, blocks of a graph

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57M15, 05C50

Milestones

Received: 16 February 2011

Revised: 29 August 2011

Accepted: 10 October 2011

Published: 10 April 2012