Combinatorial knot Floer homology and cyclic double branched covers
  Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $\Sigma_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$. 
Fatemeh Douroudian
math.GT