Instanton and pillowcase homology of the (−2,3,q) pretzel knots
Abstract
Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot T(3,5) by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots T(3,n), we compute the Floer complex and its ℤ/4 grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank q+2 for every odd q≥3, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.
Related
- Explicit formula for the Rasmussen invariant of odd pretzel knots
- Exceptional pretzel knots are not algebraically slice
- Equivariant Instanton Homology
- Traceless SU(2) characters and ℤ/4 instanton gradings for two-bridge and (3,n)-torus knots
- Fox-Milnor condition for concordant knots in homology 3-spheres
BibTeX
@misc{wuebben2026instanton,
title = {{Instanton and pillowcase homology of the $(-2,3,q)$ pretzel knots}},
author = {Bernd Johannes Wuebben},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.26096},
eprint = {2607.26096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26096},
}
RIS
TY - UNPB
TI - Instanton and pillowcase homology of the (−2,3,q) pretzel knots
AU - Bernd Johannes Wuebben
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.26096
UR - https://arxiv.org/abs/2607.26096
AB - Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot T(3,5) by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots T(3,n), we compute the Floer complex and its ℤ/4 grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank q+2 for every odd q≥3, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.
ER -