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Holomorphic Triangles in the Heegaard Floer Pillowcase

  • Eaman Eftekhary
  • Marzieh Souri
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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 1/0,p/q and 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.

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Cite

BibTeXRIS
BibTeX
@misc{eftekhary2026holomorphic,
  title = {{Holomorphic Triangles in the Heegaard Floer Pillowcase}},
  author = {Eaman Eftekhary and Marzieh Souri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21153},
  eprint = {2610.21153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21153},
}
RIS
TY  - UNPB
TI  - Holomorphic Triangles in the Heegaard Floer Pillowcase
AU  - Eaman Eftekhary
AU  - Marzieh Souri
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21153
UR  - https://arxiv.org/abs/2610.21153
AB  - 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 1/0,p/q and 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.
ER  -