Open Collections

Description

Knots and links play an important role in 3-manifolds and the equiva- lence relation of concordance of knots and links plays an important role in 4-manifolds. We will discuss our work that shows, loosely speaking, that we cannot hope to classify knot concordance without simultaneously classifying link concordance for links of an arbitrary number of components. Cochran- Friedl-Teichner considered generalized satellite operations \( R: SL(m)\to AS \), called “infection by a string link”, where \( SL(m) \) is the set of concordance classes of m-component links, \( AS\) is the set of concordance classes of algebraically slice knots, and the “pattern” knot R is some ribbon knot R. They proved that, for any such knot K there exists some \( R\), \( m\) and \( L\) such that \( R(L)=K\). We show that one cannot put an upper bound on \( m\). Links arise from knots since the spine of a Seifert surface is essentially a link. Our obstructions are related to the Alexander polynomials of such links.