Orientations for skein-valued curve counting

@misc{parker2026orientations,
  title = {{Orientations for skein-valued curve counting}},
  author = {Brett Parker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41233},
  eprint = {2610.41233},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41233},
}

TY  - UNPB
TI  - Orientations for skein-valued curve counting
AU  - Brett Parker
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41233
UR  - https://arxiv.org/abs/2610.41233
AB  - 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.
ER  -