Goeritz and Seifert matrices from Dehn presentations (Q782386)

Given a finite group presentation, Fox's free differential calculus defines a partial derivative for each relator with respect to each generator. These can then be combined into a Jacobian matrix \(J\) for the group presentation. Further matrices can be derived from \(J\) by mapping each element by a homomorphism. From a non-split link diagram, the authors construct a group presentation, and they show directly that this presentation is equivalent to a Dehn presentation of the fundamental group of the link complement, but may use far fewer generators. The Goeritz matrix of a link diagram is an integer matrix from which link invariants can be calculated. The authors' group presentation is such that, for a non-split diagram, one of the matrices derived from the Jacobian matrix is the Goeritz matrix of the diagram with an extra column of zeros. Previous descriptions of the Goeritz matrix have not worked directly from a presentation of the link group. Having established this result, the authors then extend it to split link diagrams. The final section of the paper studies Seifert matrices for special link diagrams. For an oriented link, a diagram of the link is called special if a checkerboard surface from the diagram can be oriented in keeping with the link orientation (that is, it is a Seifert surface for the link). Every link has a special link diagram. A Seifert matrix for the surface can be calculated using linking numbers of (push-offs of) loops on the surface. The authors show that, if the diagram is non-split, the group presentation they constructed can be used to construct a Seifert matrix for the checkerboard Seifert surface. If the diagram is also alternating, the group presentation is that of an HNN extension.