Skein traces from curve counting

TL;DR

This work constructs a skein-valued framework for counting holomorphic curves with Lagrangian boundary in cotangent bundles and uses this to produce skein-valued wall-crossing formulas. By relating holomorphic curves to Morse flow graphs, the authors obtain explicit skein traces for branched double covers, including a degree-2 explicit description aligning with Neitzke–Yan prescriptions. They develop a comprehensive machinery: a skein dilogarithm, branched-cover skein modules (singular and smoothed), 4-chains and brane data, and a flow-graph toolkit that links crossing changes to skein relations. The results unify wall-crossing, Lagrangian cabling, and skein algebra via a geometric, curve-counting perspective, with multiple proofs and SFT interpretations. The framework recovers key known formulas in specialized settings and provides a path to broader skein-valued DT/cluster-type phenomena in 3-manifold/topological-quantum field theory contexts.

Abstract

Given a 3-manifold $M$, and a branched cover arising from the projection of a Lagrangian 3-manifold $L$ in the cotangent bundle of $M$ to the zero-section, we define a map from the skein of $M$ to the skein of $L$, via the skein-valued counting of holomorphic curves. When $M$ and $L$ 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 $M$ is a surface times an interval, and additionally specializing the HOMFLYPT skein to the $\mathfrak{gl}(2)$ skein on $M$ and the $\mathfrak{gl}(1)$ skein on $L$, we recover an existing prescription of Neitzke and Yan.

Figures (44)

• Figure 1: Red tangle associated to the marked blue edges.
• Figure 2: Boundaries of slicing quadrilaterals.
• Figure 3: The branch locus for the singular branched double cover of a tetrahedron
• Figure 4: 3d spectral network on a tetrahedron (twelve green triangles).
• Figure 5: Leaf space / branch cuts for the branched cover (six orange quadrilaterals).

Theorems & Definitions (150)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['explicit description']}
• Theorem 1.5
• proof
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 3.1