work was done. I also thank Chuck Livingston for his interest, and Cornelia Van Cott for informing me of Kawauchi’s results and suggesting Corollary \[cor:concordance\].

Proof of Theorems {#sec:proof}
=================

In this section we prove the theorems stated in the introduction. Several of the proofs rely heavily on the results of [@Cabling].

Notational Background
---------------------

Before beginning, we recall a few facts about knot Floer homology. Our purpose here is to establish notation and is not intended as an introduction to  theory or knot Floer homology. First, recall that to a knot $K\subset Y$ we can associate a doubly-pointed Heegaard diagram $$(\Sigma,\{\alpha_1,\ldots,\alpha_g\},\{\beta_1,\ldots,\beta_{g} \},w,z).$$ (see Definition $2.4$ of [@Knots]). The  chain complex $\CFa(Y)$ is generated (as a $\Z/2\Z$ vector space) by $g$-tuples $\x=x_1\times ... \times x_g$ of intersection points, where $x_i\in \alpha_i\cap \beta_{\sigma(i)}$ (here, $\sigma$ is a permutation in the symmetric group on $g$ letters). The chain complex is equipped with a differential $\partial$ which counts points in moduli spaces of pseudo-holomorphic disks [@HolDisk].

Let $\mathcal{G}$ be the set of generators. Using the basepoint, $w$,  define a map: $$\spinc_w: \mathcal{G}\rightarrow \SpinC(Y),$$ where $\SpinC(Y)$ is the set of $\SpinC$ structures on $Y$. The chain complex splits as direct sum $$\CFa(Y)\cong \underset{\spinc\in \SpinC(Y)}\bigoplus 