A note on the knot Floer homology of fibered knots
In [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)], \textit{P. Ozsváth} and \textit{Z. Szabó} introduced some knot Floer invariants. If \(K\) is a knot in a closed oriented 3-manifold \(Y\), then one of the most basic versions of these invariants assigns to a knot \(K\) a vector space \(\widehat{HFK}(Y,K)\) over the field \(\mathbb F=\mathbb Z/2\,\mathbb Z\). If \(K\) is nullhomologous with Seifert surface \(\Sigma\), then this vector space may be endowed with an Alexander grading \(\widehat{HFK}(Y,K)=\bigoplus\limits^{g(\Sigma)}_{i=-g(\Sigma)}\widehat {HFK}(Y,K,[\Sigma],i)\) which depends only on the relative homology class of the surface in \(H_2(Y,K)\). Much of the power of knot Floer homology owes to its relationship with the genus and fiberedness of knots. For instance, if \(K\subset Y\) is fibered with fiber \(\Sigma\), then \(\widehat {HFK}(Y,K,[\Sigma],g(\Sigma))\cong\mathbb F\). This result in some way deals with the summand of knot Floer homology in the top Alexander grading. In this paper, the authors show that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. It is proven that if \(K\subset Y\) is genus \(g >0\) fibered knot with fiber \(\Sigma\), then the summand \(\widehat {HFK}(Y,K,[\Sigma],g-1)\) is nonzero. The authors present several applications of this result. If \(Y\) is a rational homology 3-sphere, then the Heegaard Floer homology of \(Y\) is bounded in rank by the number of elements in the first homology of \(Y\), \(\widehat{HF}(Y)\geq |H_1(Y)|\). An \(L\)-space is a rational homology 3-sphere \(Y\) for which this inequality is sharp \(\widehat{HF}(Y)=|H_1(Y)|\). A knot \(K\subset S^3\) is called an \(L\)-space knot if some Dehn surgery on \(K\) is an \(L\)-space. The authors show that if \(K\) is an \(L\)-space knot, then \(K\) is prime and \(\widehat {HFK}(S^3,K,g(K)-1)\cong\mathbb F\). Furthermore, the authors also introduce a numerical refinement of the Ozsváth-Szabó contact invariant. The contact invariant in Heegaard Floer homology assigns to a contact structure \(\xi\) on \(Y\) a class \(c(\xi)\in\widehat{HF}(-Y)\) which is defined as \(c(\xi)=[\mathbf c]\in H_*(\widehat{CF}(-Y),\partial)=\widehat{HF}(-Y)\).
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