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A combinatorial description of knot Floer homology. (English) Zbl 1179.57022

Knot Floer homology is a powerful knot invariant defined by Ozsváth and Szabó, and generalizes the Alexander-Conway polynomial. It was originally defined using a filtered chain complex whose differential counted pseudo-holomorphic disks, and hence it was not possible to compute it algorithmically.
This fundamental paper presents an algorithm for computing both the hat and the minus versions of knot Floer homology of a given knot \(K\). It uses a grid presentation of \(K\), from which one can construct a multi-pointed Heegaard diagram for \(K\). The Heegaard surface is a torus, and the \(\alpha\) curves are all meridians and the \(\beta\) curves are all longitudes. If the number of \(\alpha\) curves is \(n\), then there are exactly \(2n\) marked points that specify \(K\). Every pseudo-holomorphic disk can be shown to correspond to a rectangle in this grid. The authors give combinatorial formulas for the Alexander and Maslov gradings of the generators of the knot Floer chain complex, and the differential can be computed by counting empty rectangles in the grid. Even though this method is algorithmic, it is by no means efficient.
The concordance invariant \(\tau\) and link Floer homology can also be computed algorithmically from this picture. It is worth noting that this is a continuation of the work of Sarkar and Wang, who gave an algorithm for computing the hat version of Heegaard Floer homology of closed 3-manifolds.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R58 Floer homology
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