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

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 \(n\equiv1,2\pmod 6\) and exactly one, to the generator of the abelian representation, when \(n\equiv4,5\pmod 6\). 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.