A surgery formula for knot Floer homology. (English) Zbl 07828336
Summary: Let \(K\) be a rationally null-homologous knot in a \(3\)-manifold \(Y\), equipped with a non-zero framing \(\lambda\), and let \(Y_{\lambda} (K)\) denote the result of \(\lambda\)-framed surgery on \(Y\). Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of \(Y_{\lambda}(K)\) in terms of the knot Floer complex of \((Y,K)\). We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot \(K_{\lambda}\) in \(Y_{\lambda}\), i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest.
MSC:
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
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