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.

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) Reshetikhin-Turaev Invariants of Ribbon Graphs.
  
• September 29*: (Carlos Alvarado) WRT Invariants I.
  
• October 6*: WRT Invariants II.
  
• October 13*: Webs, Foams, and Khovanov Homology.
  
• October 22: Functoriality of Khovanov Homology.
  
• October 29: (Fan Zhou) Triply graded homology
  
• November 5:
• November 12:
• November 19:
• November 26: No Meeting (Thanksgiving)
• December 3:

Possible Talks:

• The Jones Polymomial:
  • Jones original construction using von Neumann algebras [Jon85,Jon91].
  • (Felix) The Kauffman bracket and the diagrammatic Temperley-Lieb algebra [Kau86].
  • (Ross) The HOMFLY-PT polynomial using Hecke algebras [Jon87].
  • (Lisa) Reshetikhin-Turaev's construction using quantum groups [RT90].
  • (Carlos) The Witten-Reshetikhin-Turaev invariants of 3-manifolds [RT91, KM91, Li97, Tu16].
• Khovanov-Rozansky homology:
  • (Felix) gl_N homology (KR_N) using webs and foams [MOY98, RW20].
  • (Fan) Triply-graded homology (HHH) using Soergel bimodules [Kho07].
• Other topics:
  • Tangle invariants and arc algebras [Kho02].
  • Matrix factorizations [KR08].
  • (Fan/Elise) Skew-Howe duality and categorified quantum groups [CKM14,LQR15,Web13].
  • (Fan/Elise) Lie theory and category O [Str05,Sus07,MS09].
  • (Alekos/Shijie?) Symplectic geometry of nilpotent slices [SS06,Man06,AS15].
  • (Alekos/Shijie?) Derived categories of coherent sheaves [CK08].
  • Stable homotopy refinements [LS14,ELST16,HZS18].
  • Higher categories of Soergel bimodules [LMRSW24, SW24].
  • Hopfological Algebra [Kho14,Qi14].
  • Witten's physical interpretations [Wit89, Wit11].

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.
• [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.