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arXiv · 2610.41233

Orientations for skein-valued curve counting

Brett Parker
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Abstract

Ekholm and Shende's skeins-on-branes formalism provides invariant counts of holomorphic curves with boundary on a Maslov zero Lagrangian within a Calabi-Yau 3-fold. This paper proves the orientation relations required for worldsheet skein relations to cancel codimension-one boundary strata. However, standard Cauchy-Riemann Fredholm orientations reveal an unexpected sign puzzle: the orientation relation at a curve with a hyperbolic boundary crossing depends on whether the crossing lies on distinct boundary components or the same component. This domain-dependent sign contradicts the uniform crossing relations of the HOMFLYPT skein algebra. We show that this discrepancy is resolved by passing to an anti-holomorphic orientation convention. We establish all required orientation comparisons in the general setting of an oriented Lagrangian submanifold of a 6-dimensional symplectic manifold.

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Brett Parker. 2026-10-30. Orientations for skein-valued curve counting. https://arxiv.org/abs/2610.41233

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