Holomorphic Triangles in the Heegaard Floer Pillowcase

Abstract

The Heegaard Floer pillowcase is a recurring local model for rational tangle replacement and exact-triangle constructions. We compute the triangle product for its standard rational-slope triples of curves and encode all contributing classes in a finite generating polynomial. One slope ordering gives a vanishing map. For the other, a closed form for the generating function is computed. One may assume that the slopes are $\tfrac10,\tfrac pq$ and $\tfrac{p'}{q'}$. When an integer lies between the two finite slopes, the closed form has a particularly simple description and there are exactly $2(pq'-p'q)+2qq'$ contributing classes. We also compute the absolute Maslov bi-gradings of the intersection generators in the employed model, relating the planar labels to the graded Floer groups of two-bridge knots and links.