On Knots with trivial Alexander polynomial
By Stavros Garoufalidis, P. Teichner. 2002
46 citations recorded by the source corpus.
We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which do not have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topology. Our examples contradict a lemma of Mike Freedman, and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b331bfc5c3d49205c0706499fbd5d5089a6f48d1
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In Discrete Mathematics and Combinatorics
- A. Stipsicz
- Peter S. Ozsv'ath
- Zolt'an Szab'o
- Ciprian Manolescu
- Jacob Rasmussen
- P. Lisca
- John A. Baldwin
- M. Hedden
In Geometry and Topology
- A. Stipsicz
- Peter S. Ozsv'ath
- Zolt'an Szab'o
- Ciprian Manolescu
- P. Lisca
- Jacob Rasmussen
- John A. Baldwin
- M. Hedden
In Mathematical Physics
- A. Stipsicz
- Peter S. Ozsv'ath
- Zolt'an Szab'o
- Ciprian Manolescu
- Jacob Rasmussen
- M. Hedden
- John A. Baldwin
- Ian Zemke
Citations in the corpus
- A rational noncommutative invariant of boundary links (2001, result): “ant settles a question of low-dimensional topology (namely, to separate minimal rank knots from Alexander polynomial 1 knots), and xes an error in earlier work of M. Freedman; see [ 26 ].”
- Grid Homology for Knots and Links (2015, background): “The fact that these knots are topologically slice follows from a famous result of Freedman [59] (see also [67]), showing that any knot whose Alexander polynomial ΔK(t) = 1 is topologically slice.”