Lagrangians for the Gopakumar-Vafa conjecture
In context of Gopakumar-Vafa's conjecture the author constructs a Lagrangian in the vector bundle \( O(-1) \oplus O(-1) \) over \( \mathbb{P} ^1 \) from a knot in \( \mathbb{R}P ^3 \). The material is divided up in 5 sections as follows. The first section describes the construction of Lagrangians in \( \mathbb{C} ^2 \) from braids in \( S ^3 \) that intersect all large radius 3-spheres as a braid that is braid isotopic to the original. Section two explains how the double points of the immersed disks relate to the crossings of a certain projection of the original braid. The third section relates the topology of three Lagrangians coming from a triad of braids that arise in standard discussions of skein relations. Section four constructs Lagrangian 3-manifolds in \( O(-1) \oplus O(-1) \) from certain Lagrangian surfaces in \( \mathbb{C} ^2 \). Finally, the last section views these \( O(-1) \oplus O(-1) \) Lagrangians from a dual perspective in \( T ^{*} S ^3 \). In conclusion, it is a very nice paper.
scientific article; zbMATH DE number 1818583