(Homological) Knot Invariants From Mirror Symmetry

Quantum invariants of knots (also known as Chern-Simons knot invariants) have many applications in mathematics and physics. For example, Khovanov showed in 1999 that the simplest such invariant, the Jones polynomial, arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: What are they homologies of?

Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to the fore. Typically, though, it leads to invariants that have no particular interest outside of the problem at hand.

In this lecture, Mina Aganagic will present how she recently showed there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants. In particular, they solve the knot categorification problem.

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