# zbMATH — the first resource for mathematics

Heegaard-Floer homology and string links. (English) Zbl 1168.57012
Let $$Y$$ be a closed oriented 3-manifold, and suppose that $$L$$ is a $$k$$-component oriented link in $$Y.$$ A d-base for $$L$$ is an oriented disc $$D$$ embedded in $$Y$$ that intersects each component of $$L$$ exactly once in a positive interior point. If $$Y = S^3,$$ then the part of $$L$$ inside $$S^3 \setminus (D \times I)$$ gives a tangle in $$D^2 \times I$$ called a string link.
In this paper, Roberts defines a Heegaard-Floer invariant of d-based links using multi-pointed Heegaard diagrams. This generalizes the Floer homology invariant of knots defined by Ozsváth and Szabó, and independently by Rasmussen. However, it is different from the link Floer homology invariant of Ozsváth and Szabó. It is worth noting that Roberts’ invariant agrees with the sutured Floer homology of a natural sutured manifold complementary to $$D \cup L.$$ In each $$\text{Spin}^c$$ structure on $$Y,$$ the d-based link invariant possesses a relative $$\mathbb{Z}^k/\Lambda$$ grading, where $$\Lambda$$ is a lattice in $$\mathbb{Z}^k.$$ This corresponds to the decomposition of sutured Floer homology along relative $$\text{Spin}^c$$ structures.
In the case of a string link $$S,$$ this invariant is a $$\mathbb{Z}^k$$ graded Abelian group that categorifies the Reidemeister torsion of $$S.$$ If $$S$$ is alternating, then the Floer homology of $$S$$ is determined by the torsion. Given a projection of $$S,$$ it is easy to draw a Heegaard diagram defining $$S,$$ where the generators correspond to certain Kauffman states. Roberts proves Künneth-type formulas for three kinds of operations on string links: placing two links side-by-side, stacking one string link on top of another if the number of strands coincide, and taking satellites. Finally, he shows a skein exact sequence for a certain class of tangles.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
Full Text:
##### References:
 [1] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5 (2003) · Zbl 1009.57003 [2] P M Gilmer, R A Litherland, The duality conjecture in formal knot theory, Osaka J. Math. 23 (1986) 229 · Zbl 0591.57001 [3] R E Gompf, A I Stipsicz, $$4$$-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999) · Zbl 0933.57020 [4] N Habegger, X S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389 · Zbl 0704.57016 [5] L H Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press (1983) · Zbl 0537.57002 [6] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987) · Zbl 0627.57002 [7] P Kirk, C Livingston, Z Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001) 87 · Zbl 0989.57005 [8] R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955 · Zbl 1130.57035 [9] R Litherland, The Alexander module of a knotted theta-curve, Math. Proc. Cambridge Philos. Soc. 106 (1989) 95 · Zbl 0705.57005 [10] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225 · Zbl 1130.57303 [11] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $$(2)$$ 159 (2004) 1027 · Zbl 1073.57009 [12] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $$(2)$$ 159 (2004) 1159 · Zbl 1081.57013 [13] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 · Zbl 1062.57019 [14] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311 · Zbl 1056.57020 [15] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326 · Zbl 1099.53058 [16] P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615 · Zbl 1144.57011 [17] J Rasmussen, Floer Homology and Knot Complements, PhD thesis, Harvard University (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.