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New Concordance Classes From Infection By A String Link

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; General topology; Algebraic topology

video/mp4

Author affiliation: University of Victoria

2016-10-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59587

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/