On the Goertiz matrix of a link
Elaborating on a result of \textit{C. McA. Gordon} and \textit{R. A. Litherland} [Invent. Math. 47, 53-69 (1978; Zbl 0391.57004)], it is shown that the Goeritz matrix associated to a diagram of an oriented classical link \(\ell\) can be modified, by adjoining a number of entries (0 and \(\pm\) 1) on the diagonal, in such a way that (up to integral congruence) the effect of the Reidemeister moves on the resulting matrix H consists solely of the adjunction and deletion of diagonal blocks \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) If \(\Lambda ={\mathbb{Z}}[t_ 1,...,t_ m,t_ 1^{- 1},...,t_ m^{-1}]\) (where m is the multiplicity of \(\ell)\) and \(\tau\) : \(\Lambda\to {\mathbb{Z}}\) maps each \(t_ i\) to -1, the elementary ideals of H are given by \(\tau E_ k(\ell)=E_{k-1}(H)\). For \(k\in {\mathbb{Z}}\) let \(\delta_ k(\ell)=\sup \{\delta \geq 0| 2^{\delta}\quad divideseveryelementof\tau E_ k(\ell)\}\) (with \(\delta_ k(\ell)=\infty\) if \(\tau E_ k(\ell)=0)\); the sequence \((\delta_ k(\ell))\) is shown to be an invariant of the topological concordance class of the unoriented link obtained from \(\ell\), and an example is given of an infinite family of two-component links which can be distinguished by their \(\delta_ 1\) but not by their Murasugi nullities, Murasugi signatures, or Milnor \({\bar \mu}\)-invariants.
Goeritz matrix of an oriented classical link
scientific article; zbMATH DE number 3852028
scientific article