Absolute $\mathbb{Z}/2$ gradings in real Heegaard Floer homology

Abstract

Real Heegaard Floer homology is an invariant associated to a three-manifold equipped with an involution with nonempty fixed set of codimension two. We show that when the image of the fixed point set is nullhomologous in the quotient, the real Heegaard Floer homology groups admit an absolute $\mathbb{Z}/2$ grading; in particular this applies to double branched covers of links in $S^3$. As an application, we define a $\mathbb{Z}$-valued invariant of knots, which is the appropriate signed analogue of Miyazawa's degree invariant. Furthermore, we show that this invariant is equal to the Alexander polynomial of the knot evaluated at $i$.