nLab Khovanov homology

Idea

Khovanov homology is a knot invariant that is a categorification of the Jones polynomial.

It is a member of the family of Khovanov-Rozansky link homology theories which categorify the HOMFLY-PT polynomial and some of its one-variable specializations.

Interpretation in QFT

Khovanov homology has long been expected to appear as the quantum observables in D=4 TQFT in higher analogy of how the Jones polynomial arises as a quantum observable in 3-dimensional Chern-Simons D=3 TQFT. For instance for $\Sigma \colon K \to K'$ a knot cobordism between two knots there is a natural morphism

$$\Phi_\Sigma \colon \mathcal{K}(K) \to \mathcal{K}(K')$$

between the Khovanov homologies associated to the two knots.

Witten 2011 argued, following indications in Gukov, Schwarz & Vafa 2005, that this 4d TQFT is related to the worldvolume theory of the image in type IIB string theory of D3-branes ending on NS5-branes in a type II supergravity background of the form $\mathbb{R}^9 \times S^1$ with the circle transverse to both kinds of branes, under one S-duality and one T-duality operation

$$(D3-NS5) \stackrel{S}{\mapsto} (D3-D5) \stackrel{T}{\mapsto} (D4-D6) \,.$$

To go from the Jones polynomial to Khovanov homology, one interprets the circle as Euclidean time. The path integral with the circle is the partition function (Witten index), $Tr_{\mathcal{H}}(-1)^F e^{-\beta H}$, of a 5D theory. Khovanov homology is $\mathcal{H}$ itself, rather than the index.

(See Witten 2011, p. 14).

Earlier indication for this had come from the observation Witten92 that Chern-Simons theory is the effective background theory for the A-model 2d TCFT (see TCFT – Worldsheet and effective background theories for details).

Related concepts

• link cobordism

• categorification in representation theory

References

General

Exposition:

• Andrew Lobb: A feeling for Khovanov homology, Notices of the AMS 71 5 (2024) [doi:10.1090/noti2928, pdf, pdf, full issue:pdf]

Original articles:

• Louis Crane, Igor Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994) 5136-5154, hep-th/9405183

• Igor Frenkel, Mikhail Khovanov, Canonical bases in tensor products and graphical calculus for $U_q(\mathfrak{sl}_2)$, Duke Math. J. 87 (1997) 409-480, MR99a:17019, doi

• Joseph Bernstein, Igor Frenkel, Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}-2)$ by projective and Zuckerman functors, Selecta. Math. 5 (1999) 199-241, MR2000i:17009, doi

• Igor Frenkel, Mikhail Khovanov, Catharina Stroppel, A categorification of finite-dimensional irreducible representations of quantum $\mathfrak{sl}_2$ and their tensor products, Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431, MR2008a:17014, doi

• Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359-426 [arXiv:math/9908171, doi:10.1215/S0012-7094-00-10131-7, MR1740682 (2002j:57025)]

• Mikhail Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665-741 MR1928174 (2004d:57016)

• Mikhail Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003) 365–374

• Magnus Jacobsson: An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211-1251 [arXiv:math/0206303, doi:10.2140/agt.2004.4.1211]

• Mikhail Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), no. 1, 315–327, arXiv:math.QA/0207264, MR2006g:57046, doi

• Raphaël Rouquier, Khovanov-Rozansky homology and 2-braid groups, arxiv/1203.5065

• Carlo Collari, The Functoriality of Khovanov Homology and the Monodromy of Knots (2013) [pdf, pdf]

Review and lecture notes:

• Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Alg. Geom. Topology 2 (2002) 337-370 [arXiv:math.GT/0201043]

• Mikhail Khovanov (notes by You Qi), Introduction to categorification, lecture notes, Columbia University (2010, 2020) [web, web, full:pdf]

• Paul Turner, Five Lectures on Khovanov Homology, [math.GT/0606464]

• Mikhail Khovanov, Robert Lipshitz, Categorical lifting of the Jones polynomial: a survey, Bulletin (New Series) of the American Mathematical Society (2022) [doi:10.1090/bull/1772]

• Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499, MR2006g:57017

See also

• David Clark, Scott Morrison, Kevin Walker, Fixing the functoriality of Khovanov homology (arXiv:math/0701339 pdf slides)

Discussion via a cobordism category of “foams”:

• Mikhail Khovanov, $sl(3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045-1081 [arXiv:math/0304375, doi:10.2140/agt.2004.4.1045]

• Marco Mackaay, Pedro Vaz, The universal $sl_3$-link homology, Algebr. Geom. Topol. 7 (2007) 1135-1169 [doi:10.2140/agt.2007.7.1135]

• Marco Mackaay, Pedro Vaz, The foam and the matrix factorization $sl_3$ link homologies are equivalent, Algebr. Geom. Topol. 8 (2008) 309-342 [arXiv:0710.0771, doi:10.2140/agt.2008.8.309]

• Marco Mackaay, Pedro Vaz, The diagrammatic Soergel category and $sl(N)$-foams, for $N \gt 3$ [arXiv:0911.2485]

• Mikhail Khovanov, Louis-Hadrien Robert, Foam evaluation and Kronheimer–Mrowka theories, Advances in Mathematics 376 (2021) 107433 [arXiv:1808.09662, doi:10.1016/j.aim.2020.107433]

• Mikhail Khovanov, Louis-Hadrien Robert, Conical $SL(3)$ foams, Journal of Combinatorial Algebra 6 1/2 (2022) 79-108 [arXiv:2011.11077, doi:10.4171/jca/61]

Reviewed in:

• Mikhail Khovanov, Universal construction, foams and link homology, lecture series at QFT and Cobordism, CQTS (Mar 2023) [web, video 1:YT, 2:YT]

A proposal to incorporate the geometry of knots to Khovanov homology:

• Li Shen, Jian Liu, Guo-Wei Wei. Evolutionary Khovanov homology (2024) [arXiv:2406.02821]

See also:

• Khaled Qazaqzeh, Nafaa Chbili, On Khovanov Homology of Quasi-Alternating Links, Mediterr. J. Math. 19 104 (2022) [arXiv:2009.08624, doi:10.1007/s00009-022-02006-5]

• Nafaa Chbili, Quasi-alternating links, Examples and obstructions, talk at QFT and Cobordism, CQTS (Mar 2023) [web]

Knot invariants via topological strings and 5-branes

On realization of knot invariants/knot homology via topological string theory and BPS states:

• Edward Witten, Chern-Simons gauge theory as a string theory, in: The Floer memorial volume, Progr. Math. 133, Birkhäuser (1995) 637-678 [doi:10.1007/978-3-0348-9217-9, arXiv/hep-th/9207094, MR97j:57052]

• Hirosi Ooguri, Cumrun Vafa: Knot Invariants and Topological Strings, Nucl. Phys. B 577 (2000) 419-438 [doi:10.1016/S0550-3213(00)00118-8, arXiv:hep-th/9912123]

• Sergei Gukov, Albert Schwarz, Cumrun Vafa: Khovanov-Rozansky Homology and Topological Strings, Lett. Math. Phys. 74 (2005) 53-74 [doi:10.1007/s11005-005-0008-8arXiv:hep-th/0412243]

• Sergei Gukov: Surface Operators and Knot Homologies, Fortschritte der Physik 55 5-7 (2007) 473-490 [doi:10.1002/prop.200610385, arXiv:0706.2369]

• Mina Aganagic, Cumrun Vafa, Large $N$ duality, mirror symmetry, and a Q-deformed A-polynomial for knots [arXiv:1204.4709]

• Kai Wang, Shengmao Zhu: BPS invariants from framed links [arXiv:2502.16609]

Understanding this via NS5-branes/M5-branes:

• Edward Witten, Fivebranes and Knots, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 [arXiv:1101.3216, doi:10.4171/QT/26]

• Davide Gaiotto, Edward Witten, Knot Invariants from Four-Dimensional Gauge Theory, Advances in Theoretical and Mathematical Physics 16 3 (2012) [doi:10.4310/ATMP.2012.v16.n3.a5, arxiv:1106.4789]

• Edward Witten: Khovanov Homology And Gauge Theory, Geometry & Topology Monographs 18 (2012) 291-308 [pdf, arXiv:1108.3103]

• Sergei Gukov, Marko Stošić: Homological algebra of knots and BPS states, Geometry & Topology Monographs 18 (2012) 309-367 [doi:10.2140/gtm.2012.18.309, arXiv:1112.0030]

Review:

• Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford (October 2013) $[$pdf]

• Ross Elliot, Sergei Gukov: Section 1 of: Exceptional knot homology, Journal of Knot Theory and Its Ramifications 25 03 (2016) 1640003 [doi:10.1142/S0218216516400034, arXiv:1505.01635]

• Satoshi Nawata, Alexei Oblomkov: Lectures on knot homology, in: Physics and Mathematics of Link Homology, Contemp. Math. 680 (2016) 137 [doi:10.1090/conm/680, arXiv:1510.01795]

An alternative approach:

• Márk Mezei, Silviu S. Pufu, Yifan Wang: Chern-Simons theory from M5-branes and calibrated M2-branes, J. High Energ. Phys. 2019 165 (2019) [doi:10.1007/JHEP08(2019)165, arXiv:1812.07572]

Related $n$Café discussions: categorification in Glasgow, Kamnitzer on categorifying tangles, link homology in Paris, 4d QFT and Khovanov homology

Parts of the above remarks on the QFT interpretation makes use of comments provided by Jacques Distler in this blog discussion.

More on relation to topological quantum field theory:

• Nitu Kitchloo, Symmetry Breaking and Link Homologies I [arXiv:1910.07443]

• Nitu Kitchloo, Symmetry Breaking and Link Homologies II [arXiv:1910.07444]

• Nitu Kitchloo, Symmetry Breaking and Link Homologies III [arXiv:1910.07516]

• Catharina Stroppel, Categorification: tangle invariants and TQFTs [arXiv:2207.05139]

Relation to gauge theory:

• Nitu Kitchloo, Symmetry breaking and homotopy types for link homologies, talk at QFT and Cobordism, CQTS (Mar 2023) [web, video:YT]