Geodesic
Heegaard–Floer homology and string links
Roberts, Lawrence1
1Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 29-101
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We extend knot Floer homology to string links in D2 × I and to d–based links in arbitrary three manifolds. As with knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D2 × I) in terms of the torsion of the string link. Additionally, a state summation approach is described using the equivalent of Kauffman states. Furthermore, we examine the situation for braids, prove that for alternating string links the Euler characteristic determines the homology, and develop similar composition formulas and long exact sequences as in knot Floer homology.

DOI : 10.2140/agt.2009.9.29
Keywords: String links, Heegaard–Floer homology, knot Floer homology

Roberts, Lawrence  1

1 Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA 
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Roberts, Lawrence. Heegaard–Floer homology and string links. Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 29-101. doi: 10.2140/agt.2009.9.29

[1] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5 (2003)

[2] P M Gilmer, R A Litherland, The duality conjecture in formal knot theory, Osaka J. Math. 23 (1986) 229

[3] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)

[4] N Habegger, X S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389

[5] L H Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press (1983)

[6] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987)

[7] P Kirk, C Livingston, Z Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001) 87

[8] R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955

[9] R Litherland, The Alexander module of a knotted theta-curve, Math. Proc. Cambridge Philos. Soc. 106 (1989) 95

[10] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225

[11] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[12] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $(2)$ 159 (2004) 1159

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

[14] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311

[15] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326

[16] P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615

[17] J Rasmussen, Floer Homology and Knot Complements, PhD thesis, Harvard University (2003)

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