Quantum Topology Bibliography

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Jones Polynomial

  • On the Origin and Development of Subfactors and Quantum Topology. Vaughan Jones. link
  • Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. Project Euclid
  • Vaughan F. R. Jones. Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. doi
  • Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. doi
  • N. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Communications in Mathematical Physics, Comm. Math. Phys. 127(1), 1-26, (1990) doi
  • N. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. doi

Categorification and Khovanov Homology

  • . Louis Crane and Igor B. Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), no. 10, 5136–5154, Topology and Physics. doi
  • Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. arXiv
  • Mikhail Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741 arXiv
  • Mikhail Khovanov, Link homology and Frobenius extensions, Fund. Math. 190 (2006), 179–190. arXiv

Khovanov-Rozansky Homology

  • Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology I. Fund. Math., 199(1):1–91, 2008. arXiv
  • Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology II, Geometry and Topology, vol.12 (2008), 1387-1425, arXiv
  • Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, International Journal of Math., vol.18, no.8 (2007), 869-885 arXiv

Functoriality

  • Dror Bar-Natan. Khovanov’s homology for tangles and cobordisms. Geom. Topol., 9:1443–1499 (electronic), 2005 arXiv
  • David Clark, Scott Morrison, and Kevin Walker. “Fixing the func- toriality of Khovanov homology”. In: Geom. Topol. 13.3 (2009), pp. 1499–1582 arXiv
  • Taketo Sano, Fixing the functoriality of Khovanov homology: A simple approach, J. Knot Theory Rami- fications 30 (2021), no. 11, 2150074, 12k arXiv
  • Michael Ehrig, Daniel Tubbenhauer, and Paul Wedrich. Functoriality of colored link homologies. Proc. Lond. Math. Soc. (3), 117(5):996–1040, 2018 arXiv

Computations

  • On the computation of torus link homology. Ben Elias and Matthew Hogancamp link
  • Joshua Wang, Colored sl(N) homology and SU(N) representations I: the trefoil and the Hopf link. arXiv
  • Joshua Wang, The Gysin sequence and the sl(N) homology of T(2,m) Proceedings for "Frontiers in Geometry and Topology" arXiv

Topological Applications

  • E.S. Lee. An endomorphism of the Khovanov invariant. Adv. Math., 197(2):554–586, 2005. arXiv
  • J. Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. arXiv
  • Lisa Piccirillo. "The Conway knot is not slice." Ann. of Math. (2) 191 (2) 581 - 591, March 2020. arXiv.
  • Peter B. Kronheimer and Tomasz S. Mrowka. Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes ´Etudes Sci., 113:97–208, 2011. arXiv
  • Peter Ozsvath and Zoltan Szabo. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194(1):1–33, 2005. arXiv

Skein Lasagna

  • S. Morrison, K. Walker, and P. Wedrich, Invariants of 4–manifolds from Khovanov–Rozansky link homology, Geom. Topol. 26 (2022), 3367–3420. arXiv
  • C. Manolescu, K. Walker, and P. Wedrich, Skein lasagna modules and handle decompositions, Adv. Math. 425 (2023), 109071. arXiv
  • C. Manolescu and I. Neithalath, Skein lasagna modules for 2-handlebodies, J. Reine Angew. Math. 2022 (2022), no. 788, 37–76. arXiv
  • Khovanov homology and exotic 4-manifolds. Qiuyu Ren, Michael Willis. arXiv

Lie Theory

  • J. Bernstein, I. Frenkel, and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quo- tients of U(sl(2)) via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241 arXiv
  • C. Stroppel, A categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors, Duke Math J. 126, no. 3 (2005), 547–596. pdf
  • V Mazorchuk and C Stroppel, A combinatorial approach to functorial quantum slk knot invariants, Amer. J. Math. 131 (2009), no. 6, 1679–1713 arXiv
  • C. Stroppel and J. Sussan, A Lie theoretic categorification of the coloured Jones polynomial. J.Pure Appl.Algebra 226 (2022), 107043. arXiv
  • J. Sussan, Category O and sl(k) link invariants. arXiv

Higher Representation Theory

  • An introduction to diagrammatic algebra and categorified quantum sl(2). Aaron D. Lauda. arXiv.
  • H. Queffelec and D.E.V. Rose. The sln foam 2-category: a combinatorial formulation of Khovanov– Rozansky homology via categorical skew Howe duality. Adv. Math., 302:1251–1339, 2016. arXiv
  • A.D. Lauda, H. Queffelec, and D.E.V. Rose. Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m). Algebr. Geom. Topol., 15(5):2517–2608, 2015. arXiv
  • S. Cautis, J. Kamnitzer, and S. Morrison. Webs and quantum skew Howe duality. Math. Ann., 360(1-2):351–390, 2014. arXiv
  • M. Mackaay and B. Webster. Categorified skew Howe duality and comparison of knot homologies. 2015. arXiv
  • Ben Webster, Knot Invariants and Higher Representation Theory. arXiv

Foam Evaluation

  • Hitoshi Murakami, Tomotada Ohtsuki, and Shuji Yamada. Homfly polynomial via an invariant of colored plane graphs. Enseign. Math. (2), 44(3-4):325–360, 1998 doi
  • Louis-Hadrien Robert and Emmanuel Wagner. A closed formula for the evaluation of foams. Quantum Topol., 11(3):411–487, 2020. arXiv

Categorification at a Root of Unity

  • A categorification of quantum sl(2) at prime roots of unity. Ben Elias, You Qi. arXiv
  • M. Khovanov. Hopfological algebra and categorification at a root of unity: The first steps. J. Knot Theory Ramifications, 25(3):359–426, 201 arXiv
  • You Qi. Hopfological algebra. Compos. Math., 150(01):1–45, 2014. arXiv:1205.1814 arXiv
  • You Qi and Joshua Sussan. On some p-differential graded link homologies, Forum of Mathematics, Pi, 10, E26 arXiv
  • You Qi and Joshua Sussan. On some p-differential graded link homologies II Algebraic and Geometric Topology Volume 23, Issue 7, 2023, 3357-3394 arXiv
  • You Qi, Louis-Hadrien Robert, Joshua Sussan, and Emmanuel Wagner. A categorification of the colored Jones polynomial at a root of unity, 2021. arXiv

Geometric Representation Theory

  • Sabin Cautis and Joel Kamnitzer. Knot homology via derived categories of coherent sheaves. I: The sl(2)-case. Duke Math. J., 142(3):511–588, 2008 arXiv
  • Sabin Cautis and Joel Kamnitzer. Knot homology via derived categories of coherent sheaves II, sl(m) case with S. Cautis, Inventiones Mathematicae. arXiv
  • Joel Kamnitzer. The Beilinson-Drinfeld Grassmannian and symplectic knot homology. Clay Mathematics Proceedings. arXiv
  • Sabin Cautis and Joel Kamnitzer. Knot homology via derived categories of coherent sheaves IV, coloured links, Quantum Topology. arXiv

Symplectic Geometry

  • Paul Seidel and Ivan Smith. A link invariant from the symplectic geometry of nilpotent slices. Duke Math. J., 134(3):453–514, 2006 arXiv
  • Ciprian Manolescu, Nilpotent slices, Hilbert schemes, and the Jones polynomial, Duke Math. J. 132 (2006), no. 2, 311–369. arXiv
  • Mohammed Abouzaid and Ivan Smith. Khovanov homology from Floer cohomology, April 2015, Journal of the American Mathematical Society 32(1) arXiv
  • Artem Kotelskiy, Liam Watson, and Claudius Zibrowius. Immersed curves in Khovanov homology, 2019. arXiv

Homotopy Theory

  • Cohen, R.L., Jones, J.D.S., Segal, G.B. (1995). Floer’s infinite dimensional Morse theory and homotopy theory. In: Hofer, H., Taubes, C.H., Weinstein, A., Zehnder, E. (eds) The Floer Memorial Volume. Progress in Mathematics, vol 133. Birkhäuser Basel. doi
  • Robert Lipshitz and Sucharit Sarkar, A Khovanov stable homotopy type, J. Amer. Math. Soc. 27 (2014), no. 4, 983–1042 arXiv
  • Robert Lipshitz and Sucharit Sarkar, A Steenrod square on Khovanov homology, J. Topol. 7 (2014), no. 3, 817–848 arXiv
  • Brent Everitt and Paul Turner, The homotopy theory of Khovanov homology, Algebr. Geom. Topol. 14 (2014), 2747–2781 arXiv
  • Brent Everitt, Robert Lipshitz, Sucharit Sarkar and Paul Turner, Khovanov homotopy types and the Dold-Thom functor, Homology, Homotopy and Applications, Vol. 18 (2016), no. 2, 177-181 arXiv