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.

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• *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.

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• October 22: (Felix Roz) Webs, Foams, and Khovanov Homology.
  Categorification is a far reaching program which seeks to realize the 3-dimensional WRT invariants as shadows of 4-dimensional invariants by enhancing the algebra used to define them. A first step is to view the integral Jones polynomial as the Euler characteristic of a homology theory for links called Khovanov Homology. To construct Khovanov homology, I will define webs, a diagrammatic calculus for the representation theory of U_q(gl_N), and foams, a 2-categorical enrichment which are treated like cobordisms between webs. In a later talk, we will discuss the advantages of the categorified version.
• October 29: (Felix Roz) Functoriality and Topological Applications.
  Last week we constructed a bigraded module over the ring of symmetric polynomials which we called the gl_N homology of a link. This week we will justify calling it a homology module by showing that it extends to a functor on the category of links and cobordisms. We will also discuss how to specialize the symmetric parameters to get finite dimensional and numerical invariants with information about the genus of knots.
• November 5: (Susan Rutter) Immersed Curves in Khovanov homology.
  
• November 12: (Elise LePage) Projective Functors.
  
• November 19: (Fan Zhou) Category O.
  
• November 26: No Meeting (Thanksgiving)
• December 3: (Shijie Dai) Coherent Sheaves.

Possible Talks:

• Skew-Howe duality and categorified quantum groups [CKM14,LQR15,Web13].
• Symplectic geometry of nilpotent slices [SS06,Man06,AS15].
• 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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