Rational surgery exact triangles in Heegaard Floer homology

We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsváth and Szabó’s $n$- and $1\mathrm{/}n$-surgery exact triangles for positive integers $n$ and the author’s recent 2-surgery exact triangle to all positive rational slopes. The construction reduces to a combinatorial problem that involves triangle and quadrilateral counting maps in a genus 1 Heegaard diagram. The main contribution of this paper is solving this combinatorial problem, which is particularly tricky for slopes $r\mathrm{\ne }n,1\mathrm{/}n$; one key idea is to use an involution that is closely related to the ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ conjugation symmetry.