time, the value $$\label{Nestimate2}
N_Y = \underset{{\mathfrak{s}}\in \mbox{\scriptsize spin${}^c$}(Y)}\max{1\over 2}( \dim_{{\mathbb F}}\widehat{HF}(Y,{\mathfrak{s}}) - 1)$$ also suffices for our theorems (here we assume $Y$ is a rational homology sphere for convenience, c.f. Remark \[Nestimateremark\]). In general produces much tighter bounds than , though both give $N_Y = 0$ if $Y$ is an $L$-space.

If we are given more data about the knot our bound can be sharpened further. To state one such result, recall that an oriented plane field distribution on a closed oriented 3-manifold is determined up to homotopy by two pieces of data: its associated [[ ]{}]{}structure, together with a “3-dimensional invariant,” as described by Gompf [@Gompf] (ultimately this classification goes back to Pontryagin). Supposing $\xi$ to be a plane field on $Y$ whose [[ ]{}]{}structure ${\mathfrak{s}}_\xi$ has torsion first Chern class, the 3-dimensional invariant is a rational number called the Hopf invariant $h(\xi)$ (see Equation in Section \[gradedsec\] below). Now whenever a [[ ]{}]{}structure has torsion Chern class, the associated Heegaard Floer homology group carries a rational-valued grading. The reduced Floer homology groups are finite-dimensional and, in particular, can be nonzero in at most finitely many degrees. Keeping this in mind, the following theorem provides a more precise 