# Knot Homology and DAHA Seminar Fall 2021

** Last Updated:**

The goal of this seminar is to understand part of the triangle of connections between Khovanov-Rozansky homology, the double affine Hecke algebra and Hilbert schemes.

### Time and location information

Organisers: Sam DeHority, Zoe Himwich, Davis Lazowski

Time/date: Tuesdays 5pm EST

Location: TBA

Please email lazowski@math.columbia.edu if you would like to be on the mailing list.

### Plan/references

First, we will cover necessary background on Hilbert schemes, DAHA and Khovanov–Rozansky homology, following (for instance)

- Punctual Hilbert Schemes, Iarrobinio
- Double Affine Hecke Algebras, Cherednik
- Lecture Notes on Cherednik Algebras, Etingof–Ma
- On Khovanov’s categorifcation of the Jones polynomial, Bar-Natan
- Lectures on knot homology, Nawata–Oblomkov

Then, depending on attendee interest, we will hone in on either the connection between DAHA and knot homology or the connection between Hilbert schemes and knot homology. (We do not expect to have time during the seminar to treat both connections adequately).

References on knot homology and Hilbert schemes:

- Refined knot invariants and Hilbert schemes, Gorsky–Negut
- Stable pairs and the HOMFLY polynomial, Maulik
- The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link, Oblomkov–Shende

References on knot homology and DAHA:

- Torus knots and the rational DAHA, Gorsky–Oblomkov–Rasmussen–Shende

We hope that talks will illustrate any connections made with explicit computations where possible.

### Virtual attendance

This seminar will be hybrid. Details TBA If you are interested in attending virtually, please let us know.

### Notes

There are live-TeXed notes of this seminar here graciously provided by Patrick Lei.

### List of talks

Date | Talk Info |
---|---|

Tuesday Sep. 14 |
Speaker: Sebastian Haney Title: Introduction to Khovanov Homology Notes:pdf |

Tuesday Sep. 21 |
Speaker: Álvaro Martínez Title: HOMFLY-PT homology Abstract:We will briefly motivate the HOMFLY polynomial, a powerful invariant of links. Then we will motivate and construct (from scratch) HHH, the categorification of this notion by Khovanov and Rozansky. There will be explicit computations and a brief overview of the surprising conjectures connecting HHH with the geometry of Hilbert schemes. Notes: pdf |

Tuesday Sep. 28 |
Speaker: Avi Zeff Title: Hilbert Schemes Abstract: We’ll introduce Hilbert schemes, look at some examples, and prove (or, time depending, state) some properties such as representability and smoothness under certain conditions. Then we’ll specialize to the cases of the plane and plane curve singularities and explore connections to representation theory and knot invariants. |

Tuesday Oct. 5 |
Speaker: Patrick Lei Title: The HOMFLY polynomial and enumerative geometry I Abstract: We will state (again) the Oblomkov–Shende conjecture and introduce an important element in its proof: stable pairs enumerative invariants. We will define the objects involved, compute an example, and then (time-permitting) start the proof of Oblomkov–Shende, following Maulik. |

Tuesday Oct. 12 |
Speaker: Patrick Lei Title: The HOMFLY polynomial and enumerative geometry II Abstract: Continued |

Tuesday Oct. 19 |
Speaker: Che Shen Title: Braid variety and Khovanov-Rozansky homology Abstract: Given a positive braid word, there is an isomorphism between its Khovanov-Rozansky homology of top degree and the equivariant cohomology of the corresponding braid variety. I will explain some of the ideas behind the proof of this theorem. In the course of the proof, I will introduce Bott-Samelson varieties, which serve as a geometric model for Khovanov-Rozansky homology on the one hand, and gives a good compactification for braid varieties on the other hand. |

Tuesday Oct. 26 |
Speaker: Sam DeHority Title: Finite dimensional type A rational Cherednik representations Abstract: The rational Cherednik algebra of type A admits finite dimensional representations at specific values of its parameter $c$. We will describe these representations using various techniques including the KZ functor and the BGG resolution. |

Tuesday Nov. 2 |
NO TALK University holiday |

Tuesday Nov. 9 |
Doubleheader. Speaker 1: Sam DeHority Continued from Oct. 26 Speaker 2: Cailan LiAbstract: I will massage what Sam wrote until it becomes a knot, figuratively speaking. |

Tuesday Nov. 16 |
Speaker: Cailan LiTitle: Torus Knots and the Rational DAHA, II Abstract: Continuing from last time we will first decompose $L_{m/n}$ as a graded $S_n$ representation into irreducibles. Then, using the representation theory of the Hecke algebra (more accurately just $S_n$) we will prove the main result connecting Cherednik algebras to link invariants, namely that the graded Frobenius character of $L_{m/n}$ coincides with the HOMFLY polynomial of the $(m,n)$ torus knot. |

Tuesday Nov. 23 |
Speaker: Davis Lazowski Title: Hilbert schemes, Coulomb branches and DAHA Abstract: I will introduce affine Grassmannians and Coulomb branches. Then I will explain how studying generalised affine springer theory for Coulomb branches gives a DAHA action on equivariant homology of punctual Hilbert schemes for plane curve singularities. |

Tuesday Nov. 30 |
Speaker: Alvaro Martinez Title: The deal with foams Abstract: We will see how the maps between representations of (quantum) $\mathfrak{sl}_n$ can be drawn as certain trivalent graphs called webs. After expressing the $\mathfrak{sl}_n$ polynomial in terms of webs, we will categorify it using foams. Foams are certain 2-dimensional CW-complexes which can be thought of as cobordisms between webs. We will go over the novel Robert-Wagner foam evaluation, and we will use it to obtain a $\mathfrak{sl}_n$ link homology alla Khovanov homology. |

Tuesday Dec. 7 |
Speaker: Alvaro Martinez Title: The deal with foams II Abstract: We will describe special kinds of webs and foams between them, and see how they give a prerequisite-free interpretation of singular Soergel bimodules, as well as part of their Hochschild homology. After that, we will show how the same machinery can be used to describe the symmetric Khovanov-Rozansky link homology, which categorifies the RT invariant corresponding to symmetric power representations of slN. |