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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)
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