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

TL;DR

This paper establishes an absolute rac{1}{2} grading in real Heegaard Floer homology for certain 3-manifolds with involutions, leading to a new knot invariant related to the Alexander polynomial.

Contribution

It introduces an absolute rac{1}{2} grading in real Heegaard Floer homology and defines a new a3-valued knot invariant linked to the Alexander polynomial.

Findings

Real Heegaard Floer homology admits an absolute rac{1}{2} grading under specified conditions.

A new a3-valued knot invariant is defined, generalizing Miyazawa's degree invariant.

The invariant equals the Alexander polynomial evaluated at i.

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 $i$.