Seminar on Quantum Invariants of Links

Abstract: This is a seminar about invariants of links coming from representation theory and categorification. We will focus on the Jones polynomial, the WRT invariants, Khovanov--Rozansky homology, triply graded homology and various variations of these varied invariants.
Time: Wednesdays 4-5pm.
Location: Math 528.

Meeting dates marked with a * are being held on Monday at 2pm in 622

Schedule:

September 10: (Felix Roz) Organizational meeting.
Overview of the topics and assigning talks. If there's time I'll define the Jones polynomial using the Kauffman bracket and the diagrammatic Temperley-Lieb algebra.
September 17: (Ross Akhmechet) The HOMFLY-PT polynomial.
The HOMFLY-PT polynomial is a two-variable link invariant that specializes to the Jones and Alexander polynomials. I'll discuss one way to define it via the Ocneanu trace of Hecke algebras.

*September 22: (Lisa Faulkner Valiente) WRT 0: Invariants of Ribbon Graphs.
We put a braided monoidal category structure on the category of representations of a Hopf algebra and construct a functor from the category of coloured ribbon graphs to this category. This provides a link invariant and generalizes the Jones polynomial to tangles.
*September 29: (Carlos Alvarado) WRT I: The 3-manifold invariant.
Last time we saw how to use ribbon categories to obtain link invariants. These do not generally give 3-manifold invariants via surgery so we introduce modular categories to obtain one.
*October 6: (Azélie Picot) WRT II: The (2+1)-dimensional TQFT.
Last time, we constructed an invariant of three-dimensional closed oriented manifolds, the WRT invariant. In this talk, we will extend this invariant to a TQFT. Roughly speaking, this (2+1)-TQFT will assign to every surface a module (over some ring k), and will assign to every three-dimensional cobordism a map between the corresponding modules. This assignment should respect disjoint union of surfaces, as well as gluing of cobordisms along a common boundary. To do so, we will decorate 3-manifolds with coloured ribbon graphs.
*October 13: (Peter Moody) WRT III: What did Witten do?
We will physically motivate the definition of a QFT as a functor on a bordism category and explain the physical interpretation of the Jones polynomial.

October 22: (Felix Roz) Webs, Foams, and Khovanov Homology.
October 29: (Felix Roz) Functoriality of Khovanov Homology.
November 5: (Fan Zhou) Triply graded homology
November 12:
November 19:
November 26: No Meeting (Thanksgiving)
December 3:

Possible Talks:

(Fan/Elise) Skew-Howe duality and categorified quantum groups [CKM14,LQR15,Web13].
(Fan/Elise) Lie theory and category O [Str05,Sus07,MS09].
(Shijie) Symplectic geometry of nilpotent slices [SS06,Man06,AS15].
(Shijie) Derived categories of coherent sheaves [CK08].
Tangle invariants and arc algebras [Kho02].
Matrix factorizations [KR08].
Stable homotopy refinements [LS14,ELST16,HZS18].
Higher categories of Soergel bimodules [LMRSW24, SW24].
Hopfological Algebra [Kho14,Qi14].

References:

[Jon85] Vaughan Jones. A polynomial invariant for knots via von Neumann algebras. 1985.
[Kau86] Louis Kauffman. State models and the Jones polynomial. 1986.
[Jon87] Vaughan Jones. Hecke algebra representations of braid groups and link polynomials.. 1987.
[Wit89] Edward Witten. Quantum field theory and the Jones polynomial. 1989.
[RT90] Nicolai Reshetikhin and Vladimir Turaev. Ribbon graphs and their invariants derived from quantum groups. 1990.
[RT91] Nicolai Reshetikhin and Vladimir Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. 1991.
[KM91] Robion Kirby and Paul Melvin. The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). 1991.
[Jo91] Vaughan Jones. Subfactors and Knots. 1991.
[Li97] W.B.R. Lickorish. An Introduction to Knot Theory. 1997.
[MOY98] Hitoshi Murakami, Tomotada Ohtsuki, and Shuji Yamada. Homfly polynomial via an invariant of colored plane graphs. 1998.
[Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. 2000.
[Kho02] Mikhail Khovanov. A functor-valued invariant of tangles. 2002.
[Oh02] Tomotada Ohtsuki. Quantum Invariants: A Study of Knots, 3-manifolds, and Their Sets. 2002.
[Str05] Catharina Stroppel, A categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors . 2005.
[SS06] Paul Seidel and Ivan Smith. A link invariant from the symplectic geometry of nilpotent slices. 2006.
[Man06] Ciprian Manolescu. Nilpotent slices, Hilbert schemes, and the Jones polynomial. 2006.
[Kho07] Mikhail Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules. 2007.
[Sus07] Josh Sussan. Category O and sl(k) link invariants. 2007.
[CK08] Sabin Cautis and Joel Kamnitzer. Knot homology via derived categories of coherent sheaves. I: The sl(2)-case. 2008.
[KR08] Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology I. 2008.
[MS09] Volodymyr Mazorchuk and Catharina Stroppel. A combinatorial approach to functorial quantum slk knot invariants. 2009.
[Wit11] Edward Witten. Fivebranes and Knots. 2011.
[HKK12] Po Hu, Daniel Kriz, Igor Kriz. Field theories, stable homotopy theory and Khovanov homology. 2012.
[Web13] Ben Webster. Knot Invariants and Higher Representation Theory. 2013.
[LS14] Robert Lipshitz and Sucharit Sarkar. A Khovanov stable homotopy type. 2014.
[LS14b] Robert Lipshitz and Sucharit Sarkar. A Steenrod square on Khovanov homology. 2014.
[ET14] Brent Everitt and Paul Turner. The homotopy theory of Khovanov homology. 2014.
[Kho14] Mikhail Khovanov. Hopfological algebra and categorification at a root of unity. 2014.
[Qi14] You Qi. Hopfological algebra. 2014.
[CKM14] Sabin Cautis, Joel Kamnitzer, and Scott Morrison. Webs and quantum skew Howe duality. 2014.
[LQR15] Aaron Lauda, Hoel Queffelec, and David Rose. Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m). 2015.
[AS15] Mohammed Abouzaid and Ivan Smith. Khovanov Homology from Floer Cohomology. 2015.
[Tu16] Vladimir Turaev. Quantum Invariants of Knots and 3-Manifolds. 2016.
[ELST16] Brent Everitt, Robert Lipshitz, Sucharit Sarkar and Paul Turner. Khovanov homotopy types and the Dold-Thom functor. 2016.
[HZS18] Po Hu, Igor Kriz, Petr Somberg. Derived representation theory of Lie algebras and stable homotopy categorification of slk. 2018.
[RW20] Louis-Hadrien Robert and Emmanuel Wagner. A closed formula for the evaluation of foams. 2020.
[LMRSW24] Yu Leon Liu, Aaron Mazel-Gee, David Reutter, Catharina Stroppel, Paul Wedrich. A braided monoidal (inf,2)-category of Soergel bimodules. 2024.
[SW24] Catharina Stroppel, Paul Wedrich. Braiding on type A Soergel bimodules: semistrictness and naturality. 2024.

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