Instanton Floer homology and the Alexander polynomial
By P. Kronheimer, T. Mrowka. 2009
59 citations recorded by the source corpus.
The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/938bf7061cd79d27ae1d5c69667487c47bf02e33
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In Discrete Mathematics and Combinatorics
- Christopher Herald
- M. Hedden
- Paul A. Kirk
- Jacob Rasmussen
- Jennifer Hom
- David Krcatovich
- Yi Ni
- J. Greene
In Geometry and Topology
- Christopher Herald
- M. Hedden
- Paul A. Kirk
- Jacob Rasmussen
- Jennifer Hom
- David Krcatovich
- Yi Ni
- E. Gorsky
In Mathematical Physics
- Christopher Herald
- M. Hedden
- Paul A. Kirk
- Jacob Rasmussen
- Jennifer Hom
- David Krcatovich
- Yi Ni
- E. Gorsky
Citations in the corpus
- The pillowcase and perturbations of traceless representations of knot groups (2013, methodology): “In [25] they establish that I\(S3, T4,5) = (2, 1, 2, 2) using the results of [21, 28] which identify the coefficients of the Alexander polynomial with a kind of Euler characteristic associated to the generalized eigenspaces of an operator μ : KHI(K) → KHI(K).”