Flexible isotopy classification of flexible links

Björklund, Johan

doi:10.1142/s0218216516500498

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Linking Numbers Of Open Curves and Lagrangians1

Introduction1

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2019

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“…Nevertheless the special geometry, plus certain additional choices, will allow us to define this number. Similar constructions was used before in the context of real algebraic geometry, see [52,8,4].…”

Section: Linking Numbers Of Open Curves and Lagrangiansmentioning

confidence: 87%

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Skeins on Branes

Ekholm¹,

Shende²

2019

Preprint

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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.

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Section: Linking Numbers Of Open Curves and Lagrangiansmentioning

confidence: 87%

“…The cotangent bundle T * S 3 and the total space X of the bundle O(−1) ⊕ O(−1) → CP 1 are respectively a deformation and a resolution of a quadric cone in C 4 . The cone itself then corresponds respectively to S 3 of radius zero and the area of CP 1 equal to zero.…”

Section: Introductionmentioning

confidence: 99%