Knot Floer homology and rational surgeries
The article constructs a version of knot Floer homology in the context of \(K\) being a rationally null-homologous knot in a three-manifold \(Y\). Also, the Floer homology of a three-manifold obtained by Morse surgery (integer slope) on \(K\) is described. Consider \(Y_{r}(K)\) the 3-manifold obtained by surgering \(Y\) along \(K\) with slope \(r\). The Heegaard Floer homology of rational surgeries on \(Y\) along a null-homologous knot \(K\subset Y\) is expressed in terms of a filtered homotopy type of the knot invariant for \(K\). After constructing the filtration, the first theorem proved is the following Theorem 1.1. Let \(K\subset Y\) be a null-homologous knot, and let \(p\), \(q\) be a pair of relatively prime integers. Then, for each \(i\in\mathbb{Z}_{p}\), there is a relatively graded isomorphism of groups \[ H_{*}(\mathbb{X}^{+}_{i,p/q})\simeq HF^{+}(Y_{p/q}(K),i), \] where \(\mathbb{X}^{+}_{i,p/q}\) is constructed from the filtration. Next, some applications come out of theorem 1.1: Application I -- A three manifold \(Y_{r}(K)\) is an \(L\)-space if it is a rational homology 3-sphere whose Floer homology \(HF^{+}(Y_{r}(K),s)\simeq HF^{+}(S^{3},s)\). A knot \(K\subset Y\) admits an \(L\)-space surgery if there exists \(r\) such that \(Y_{r}(K)\) is an \(L\)-space. The difference among the gradings of \(\widehat{HF}(S^{3}_{p/q}(K))\) and \(\widehat{HF}(S^{3}_{p/q}(O))\) (\(O\)=trivial knot) is expressed in terms of the Alexander polynomial coefficients of \(K\); the authors prove the following result; {\textbf{ Theorem 1.2}} Let \(K\subset S^{3}\) be a knot which admits an \(L\)-space surgery, for some \(r=p/q\in\mathbb{Q}\) with \(r\geq 0\). Then, for all integers \(i\) with \(|i|\leq p/2\) we have that \[ d(S^{3}_{p/q}(K),i)-d(S^{3}_{p/q}(O,i)=-2t\mid[i/q]\mid(K), \] while for all \(|j|\geq \frac{p}{2q}\), we have that \(t_{j}(K)=0\). In Theorem 1.2, \(d\) stands for a correction term function \(d:Spin^{c}(Y)\rightarrow \mathbb{Q}\) which associates to each \(s\in Spin^{c}(Y)\) the degree in which \(\widehat{HF}(Y,s)\) is supported. The authors conclude that knot Floer homology distinguishes the unknot: Corollary 1.3. If \(S^{3}_{p/q}(K)\simeq S^{3}_{p/q}(O)\) as oriented three manifolds, then \(K=O\). Application II -- cosmetic surgeries in \(S^{3}\): Let \(K\subset Y\) be a framed knot. If there are two rational numbers \(r,s\) with the property that \(Y_{s}(K)\) is homeomorphic to \(Y_{r}(K)\), then the surgeries are called cosmetic. The authors prove the next two theorems; Theorem 1.5. If \(K\) is a knot with Seifert genus equal to one, and \(S^{3}_{r}(K)\simeq S^{3}_{s}(K)\), with \(r\neq s\), then \(S^{3}_{r}(K)\) is an \(L\)-space. Theorem 1.5 places severe restrictions on \(K\), in particular the authors claim that \(K\) must have the same knot Floer homology as the trefoil knot \(T\), and thus according to theorem 1.1, \(S^{3}_{r}(K)\) and \(S^{3}_{r}(T)\) have the same (graded) Floer homology groups. More generally, they prove Theorem 1.6. If \(K\subset S^{3}\) is a knot and suppose \(S^{3}_{r}(K)\simeq S^{3}_{s}(K)\), then either \(S^{3}_{r}(K)\) is an \(L\)-space or \(r\) and \(s\) have opposite signs. Theorems 1.5 and 1.6 yield powerful methods to exclude cosmetic surgeries, as shown by the case when \(p=3\); Theorem 1.7. Let \(p=3\) and suppose that \(K\subset S^{3}\) is a knot with the property that \(S^{3}_{3/q}(K)\simeq S^{3}_{3/q'}(K)\) as oriented manifolds, then \(q'=q\). Application III -- the Heegaard Floer homology, with \(\mathbb{Z}_{2}\) coefficients, of any Seifert fibered space whose first Betti number is even is described.
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