Pillowcase Floer homology of the torus knots T(3,n)

Abstract

Hedden, Herald and Kirk defined a Lagrangian Floer homology in the pillowcase from a tangle decomposition of a knot and conjectured that, for a suitable decomposition, it recovers reduced singular instanton homology. We compute this homology for every torus knot $T(3,n)$, using their decomposition. For all sufficiently small holonomy perturbations in an explicit closed cone, the immersed curves are restricted and the complex has $1+|\sigma |$ generators, where $\sigma$ is the signature. There is no differential when $n\equiv 1,2\left(\mathrm{mod}6\right)$ and exactly one, to the generator of the abelian representation, when $n\equiv 4,5\left(\mathrm{mod}6\right)$. The $\mathbb{Z}/4$-graded homology has the graded dimensions of reduced instanton homology. This verifies the conjecture of Hedden, Herald and Kirk for this infinite family, and, combined with a cobordism inequality of Daemi and Scaduto, shows that over ${\mathbb{F}}_2$ the pillowcase complex is S-chain homotopy equivalent to their singular instanton S-complex. The proof rests on an explicit description of the traceless character variety and the strict monotonicity of an angle function along its components; three inequalities are verified by interval arithmetic. For a different decomposition of $T(3,4)$ in the same family, a numerical computation gives homology of rank $7$ rather than $5$ with zero bounding cochain.