Concordance of links with identical Alexander invariants
Jae Choon Cha, Stefan Friedl, and Mark Powell
TL;DR
This paper demonstrates that, aside from two known exceptional cases, the Alexander invariants do not uniquely determine the concordance class of links or knots, highlighting the limitations of these invariants in classifying concordance.
Contribution
The paper proves that the Alexander invariants only determine the concordance class in two specific cases, extending previous results and showing these are the only exceptions.
Findings
Alexander invariants determine concordance class only in two known cases.
In all other cases, the concordance class is not determined by Alexander invariants.
The results clarify the limitations of Alexander invariants in link and knot concordance classification.
Abstract
J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial one. A similar result for knots with Alexander polynomial one was shown earlier by M. Freedman. We prove that these two cases are the only exceptional cases, by showing that the link concordance class is not determined by the Alexander invariants in any other case.
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