Geodesic
Instanton Floer homology and the Alexander polynomial
Kronheimer, Peter1 ; Mrowka, Tomasz2
1Department of Mathematics, Harvard University, Cambridge MA 02138
2Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139
Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1715-1738
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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.

DOI : 10.2140/agt.2010.10.1715
Keywords: knot, Alexander polynomial, instanton, Floer homology, Yang–Mills

Kronheimer, Peter  1   ; Mrowka, Tomasz  2

1 Department of Mathematics, Harvard University, Cambridge MA 02138
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139 
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Kronheimer, Peter; Mrowka, Tomasz. Instanton Floer homology and the Alexander polynomial. Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1715-1738. doi: 10.2140/agt.2010.10.1715

[1] P J Braam, S K Donaldson, Floer’s work on instanton homology, knots and surgery, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 195

[2] S K Donaldson, Floer homology groups in Yang–Mills theory, 147, Cambridge Univ. Press (2002)

[3] A Floer, Instanton homology, surgery, and knots, from: "Geometry of low-dimensional manifolds, 1 (Durham, 1989)" (editors S K Donaldson, C B Thomas), London Math. Soc. Lecture Note Ser. 150, Cambridge Univ. Press (1990) 97

[4] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429

[5] E P Klassen, Representations of knot groups in SU(2), Trans. Amer. Math. Soc. 326 (1991) 795

[6] P Kronheimer, T Mrowka, Knot homology groups from instantons

[7] P Kronheimer, T Mrowka, Knots, sutures and excision

[8] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007)

[9] Y Lim, Instanton homology and the Alexander polynomial, Proc. Amer. Math. Soc. 138 (2010) 3759

[10] V Muñoz, Ring structure of the Floer cohomology of Σ × S1, Topology 38 (1999) 517

[11] Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577

[12] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[13] J A Rasmussen, Floer homology and knot complements, ProQuest LLC (2003) 126

[14] D Rolfsen, Knots and links, 7, Publish or Perish (1990)

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