Skeins on Branes

Abstract

We give a geometric interpretation of the coefficients of the HOMFLYPT polynomial of any link in the three-sphere as counts of holomorphic curves. The curves counted live in the resolved conifold where they have boundary on a shifted copy of the link conormal, as predicted by Ooguri and Vafa [38]. To prove this, we introduce a new method to define invariant counts of holomorphic curves with Lagrangian boundary: we show geometrically that the wall crossing associated to boundary bubbling is the framed skein relation. It then follows that counting holomorphic curves by the class of their boundary in the skein of the Lagrangian brane gives a deformation invariant curve count. This is a mathematically rigorous incarnation of the fact that boundaries of open topological strings create line defects in Chern-Simons theory as described by Witten [56].The technical key to skein invariance is a new compactness result: if the Gromov limit of J-holomorphic immersions collapses a curve component, then its image has a singularity worse than a node.

Displacement energy of Lagrangian 3-spheres

“…The above theorem tells us that when the complex structure is good and there is no Hamiltonian perturbation of L, there is actually no holomorphic disks touching S representing certain classes. One essential reason is that D r T * S 3 is "positive enough" to force holomorphic curves lie outside a neighborhood of S. This fact is also proved in Section 7 of [10] by a SFT stretching argument to identify open Gromov-Witten invariants under conifold transitions. Their story is more complicated than ours since we only care about some minimal classes.…”

“…Therefore our baby theory only works modulo some energy. Recently, the open Gromov-Witten theory in T * S 3 with all genera has been successfully related to knot-theoretic invariants by Ekholm-Shende [10]. It would be interesting to try to apply the techniques therein to define a full genus Floer theory, starting with the monotone Lagrangian torus in T * S 3 .…”

Augmentations, annuli, and Alexander polynomials

“…Remark 3.6. The count of annuli can also be compared to the count of holomorphic curves in the framed skein module of the brane as in [ES19], where only so called bare curves are counted.…”

“…The first case of degeneration and gluing for annuli was called elliptic boundary splitting in [ES19], and is depicted in Figure 4. It corresponds to the limit where the modulus of the annulus converges to infinity, by having one of its boundary loops shrink to a point, and can be described as follows.…”

“…The second degeneration concerns instants when annuli without positive punctures split off of 1parameter families of annuli with one positive puncture, it was called hyperbolic boundary splitting in [ES19], see Figure 5. The limit configuration consists of a rigid annulus and a disk intersecting it at a boundary point.…”