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Algebra and geometry of link homology

Eugene Gorsky, Oscar Kivinen, José Simental

Abstract

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro-geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

Algebra and geometry of link homology

Abstract

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro-geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

Paper Structure

This paper contains 29 sections, 37 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Further reading
  4. Background on link invariants
  5. Khovanov-Rozansky homology: definitions and computations
  6. Soergel bimodules and Rouquier complexes
  7. Khovanov-Rozansky homology
  8. Recursions and parity
  9. Examples
  10. Braid varieties
  11. Braid varieties and their properties
  12. Homology of braid varieties
  13. From braid varieties to positroid varieties
  14. $y$-ification and tautological classes
  15. Polynomial action
  16. ...and 14 more sections

Key Result

Theorem 2.1

Any link can be obtained as a closure of some braid.

Theorems & Definitions (120)

  • Theorem 2.1: Alexander
  • Theorem 2.2: Markov
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Example 3.5
  • Remark 3.6
  • Remark 3.7
  • ...and 110 more