## A note on the knot Floer homology of fibered knots.(English)Zbl 1408.57012

In [Adv. Math. 186, No. 1, 58–116 (2004; Zbl 1062.57019)], P. Ozsváth and 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 $$c(\xi)=[\mathbf c]\in H_*(\widehat{CF}(-Y),\partial)=\widehat{HF}(-Y)$$.

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57R17 Symplectic and contact topology in high or arbitrary dimension 57R58 Floer homology

Zbl 1062.57019
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### References:

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