The instanton homology of the $(-2,3,q)$ pretzel knots and Maurer-Cartan deformations in the two-arc algebra of the pillowcase

Abstract

For every odd $q\ge 3$ we prove that the reduced singular instanton knot homology of the pretzel knot $P(-2,3,q)$ is free abelian of rank $q+2$, by squeezing it between Hironaka's Alexander polynomial and Manion's integral Khovanov homology through the Kronheimer-Mrowka spectral sequence. On the pillowcase side we work in the wrapped Fukaya subcategory that Cazassus-Herald-Kirk-Kotelskiy identify with twisted complexes over the two-arc algebra of Kotelskiy-Watson-Zibrowius. There the higher products vanish, so the Maurer-Cartan equation for a deformation is the finite identity $(\delta +b{)}^2=0$, and a filtration lemma proved here reduces the infinite-dimensional morphism complex between any two finite twisted complexes to three integers. Encoding the $q=7$ curves, we find that smoothing the Conway-sum curve at any one of four self-intersections yields an exact Maurer-Cartan element that raises the pairing with the earring from rank $7$ to rank $9=q+2$; the four objects fall into three homotopy classes, and pairing against a second closure of the same tangle, whose instanton rank is again computed by the first theorem, isolates one class among all $82$ single smoothings. That selection is conditional on the conjectural instanton-pillowcase correspondence and on a stated localization hypothesis. We also show that the formal route through Gao's representability theorem is closed: the Lagrangian correspondence induced by the $Q_{1/3}$ line is immersed with a triple point, hence not embedded.

24 pages, 1 figure