Skein traces from curve counting
Tobias Ekholm, Pietro Longhi, Sunghyuk Park, Vivek Shende

TL;DR
This paper introduces a skein-valued counting method for holomorphic curves in branched covers of 3-manifolds, revealing a skein-theoretic wall-crossing formula and explicit skein trace formulas, connecting to known invariants.
Contribution
It defines a new skein-valued curve counting framework for branched covers and establishes a skein lift of the wall-crossing formula, with explicit formulas in special cases.
Findings
Wall crossings obey a skein-valued Kontsevich-Soibelman formula.
Explicit skein trace formulas for branched double covers.
Connection to existing HOMFLYPT and $rak{gl}$ skein invariants.
Abstract
Given a 3-manifold , and a branched cover arising from the projection of a Lagrangian 3-manifold in the cotangent bundle of to the zero-section, we define a map from the skein of to the skein of , via the skein-valued counting of holomorphic curves. When and are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula. Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the the skein trace. After specializing to the case where is a surface times an interval, and additionally specializing the HOMFLYPT skein to the skein on and the skein on , we recover an existing prescription of Neitzke and…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Advanced Operator Algebra Research
