{ "id": "epvs8xkdx3", "slug": "mikhail-khovanov", "title": "Mikhail Khovanov", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists" }, { "id": "physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics" }, { "id": "physical.scientists.mathematics-statistics.topologists-and-geometers", "label": "Topologists and geometers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.topologists-and-geometers" }, { "id": "physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists" } ], "geo": [ { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "United States · 1946 to 2000: Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "path": [ { "id": "geo.us", "label": "United States", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us" }, { "id": "geo.us.t1946", "label": "United States · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946" }, { "id": "geo.us.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical" }, { "id": "geo.us.t1946.physical.scientists", "label": "Physical and mathematical scientists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics", "label": "Mathematicians and statisticians", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers", "label": "Topologists and geometers", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers" }, { "id": "geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "label": "Low-dimensional and knot theorists", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.us.t1946.physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists" } ] } ], "excerpt": "Mikhail Khovanov is a mathematician, formerly at Columbia and now a professor at Johns Hopkins University, who created Khovanov homology, a link invariant that categorifies the Jones polynomial and detects the unknot.", "snippet": "Mikhail Khovanov is a mathematician, formerly at Columbia and now a professor at Johns Hopkins University, who created Khovanov homology, a link invariant that categorifies the Jones polynomial and detects the unknot.", "node": "physical.scientists.mathematics-statistics.topologists-and-geometers.low-dimensional-and-knot-theorists", "markdown": "# Mikhail Khovanov\n\n**Mikhail Khovanov** is a mathematician, now Professor of Mathematics at [Johns Hopkins University](https://www.edgepedia.org/johns-hopkins-university) after a career at Columbia[1](https://sites.google.com/view/mkhovanov/), who created [Khovanov homology](https://www.edgepedia.org/khovanov-homology), a link homology theory that categorifies the [Jones polynomial](https://www.edgepedia.org/jones-polynomial), and built a research program around link homology, categorified quantum groups, and topological quantum field theory (TQFT)[2](https://www.ias.edu/scholars/mikhail-khovanov).\n\n| Key fact | Detail |\n|---|---|\n| Signature result | Khovanov homology, a bigraded link homology theory whose Euler characteristic is the Jones polynomial, announced at the end of the 20th century and published in Duke Math. J. 101 (2000), 359–426[3](https://arxiv.org/abs/math.QA/9908171) • [4](https://ar5iv.labs.arxiv.org/html/math/0605339) • [5](https://www.icts.res.in/sites/default/files/Jozef%20H.%20Przytycki%20-%20ICTS%20-%20KT%20-%2026082020.pdf) |\n| Strength | Khovanov homology contains strictly more information than the Jones polynomial: there are pairs of knots with identical Jones polynomial but distinct Khovanov homology[6](https://google.iopscience.iop.org/article/10.1088/2632-072X/adde9f) |\n| Detection | Khovanov homology detects the unknot, the unlink, the trefoils, and the Hopf links; whether the Jones polynomial detects the unknot is still open[7](https://ar5iv.labs.arxiv.org/html/1811.07848) |\n| Payoff | Jacob Rasmussen's s-invariant, built on Khovanov homology, gave a combinatorial proof of the Milnor conjecture on the slice genus of torus knots[4](https://ar5iv.labs.arxiv.org/html/math/0605339) |\n| Positions | Professor of Mathematics at Johns Hopkins University; formerly at Columbia University; listed by the Institute for Advanced Study as a scholar in geometric representation theory[8](https://math.columbia.edu/~khovanov/) • [1](https://sites.google.com/view/mkhovanov/) • [2](https://www.ias.edu/scholars/mikhail-khovanov) |\n| Influence | About 8,566 citations on Google Scholar; research areas listed as categorification, link homology, TQFT, representation theory, and low-dimensional topology[9](https://scholar.google.com/citations?user=_PCGIMkAAAAJ&hl=en) |\n\n## Early career\n\nThe documented record of Khovanov's early career is short: he received his PhD in 1997, and at the end of the 20th century announced the homology theory categorifying the Jones polynomial[5](https://www.icts.res.in/sites/default/files/Jozef%20H.%20Przytycki%20-%20ICTS%20-%20KT%20-%2026082020.pdf). The foundational paper is dedicated \"to my teacher Igor Frenkel\", and in joint work with [Joseph Bernstein](https://www.edgepedia.org/joseph-bernstein) and Frenkel Khovanov had already proposed a categorification of the representation theory underlying quantum group link invariants, the seed of the program that produced Khovanov homology[3](https://arxiv.org/abs/math.QA/9908171). The Jones polynomial itself, discovered in May 1984, had revolutionized the theory of knot and link invariants, and the Crane–Frenkel conjecture on categorifying it supplied the motivating question[5](https://www.icts.res.in/sites/default/files/Jozef%20H.%20Przytycki%20-%20ICTS%20-%20KT%20-%2026082020.pdf) • [10](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/).\n\n## Career and positions\n\nKhovanov's Columbia homepage identifies him as a professor in the Mathematics Department at Columbia University, at 2990 Broadway, New York[8](https://math.columbia.edu/~khovanov/). His personal site states that he is now a Professor of Mathematics at Johns Hopkins University, having moved there from Columbia, with an office in Krieger Hall, Baltimore[1](https://sites.google.com/view/mkhovanov/). The [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study) lists him as a scholar in Geometric Representation Theory whose work studies the interrelation between link homology, TQFTs, homological algebra, geometric representation theory, and the [Langlands program](https://www.edgepedia.org/langlands-program)[2](https://www.ias.edu/scholars/mikhail-khovanov).\n\n## Khovanov homology: the construction\n\n**Categorification.** Khovanov homology assigns to an oriented link bigraded homology groups that are functorial up to sign under smooth link cobordisms and have the Jones polynomial as their [Euler characteristic](https://www.edgepedia.org/euler-characteristic): the Jones polynomial is recovered as the alternating sum of the ranks of the homology groups, with the formula summing over the two gradings[4](https://ar5iv.labs.arxiv.org/html/math/0605339) • [10](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/). In Khovanov's own November 2023 formulation, a link cobordism S induces a map of bidegree (0, −χ(S)), where χ(S) is the Euler characteristic of the cobordism surface[11](http://media.scgp.stonybrook.edu/presentations/2023/20231107_Khovanov.pdf).\n\n**The cube of resolutions.** The construction categorifies the Kauffman bracket description of the Jones polynomial. It associates a commutative n-dimensional cube of maps to the resolutions of an n-crossing diagram and collapses the cube to a complex of graded abelian groups; Reidemeister moves lift to homotopy equivalences between the complexes, so the isomorphism class of the bigraded groups is a link invariant[4](https://ar5iv.labs.arxiv.org/html/math/0605339) • [10](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/). The homology of the unknot is the ring Z[X]/(X²)[4](https://ar5iv.labs.arxiv.org/html/math/0605339).\n\n**The 4D TQFT conjecture.** The original paper conjectures that the construction yields a 4-dimensional topological quantum field theory, restricted to links in R³ and cobordisms between them. Despite many insights into the possible structure of such a theory, its existence still remains a conjecture[3](https://arxiv.org/abs/math.QA/9908171) • [10](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/).\n\n## Rasmussen's s-invariant and the Milnor conjecture\n\nThe Milnor conjecture states that the slice genus of the (p,q)-torus knot is (p−1)(q−1)/2; it was originally proved by Kronheimer and Mrowka using gauge theory. Jacob Rasmussen used Khovanov homology and its deformation studied by E.-S. Lee to give a combinatorial proof[4](https://ar5iv.labs.arxiv.org/html/math/0605339). The mechanism is the Rasmussen invariant s(K), obtained by replacing the ground ring with Q[t] and the unknot algebra with Q[t,X]/(X² − t); s(K) is an invariant of knot concordance that gives a lower bound on the slice genus, a bound that is explicitly computable and sharp on positive knots[11](http://media.scgp.stonybrook.edu/presentations/2023/20231107_Khovanov.pdf).\n\nThe consequences reach into smooth four-dimensional topology. The s-invariant can show that some knots that are topologically slice are not smoothly slice, which implies the existence of exotic smooth structures on R⁴, and it reproves that existence via knots that are topologically but not smoothly slice[10](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/) • [11](http://media.scgp.stonybrook.edu/presentations/2023/20231107_Khovanov.pdf). Later, a computer calculation of Khovanov-type invariants helped prove that the Conway knot is not smoothly slice, and a possible, yet to be realized, attack on the smooth 4-dimensional [Poincaré conjecture](https://www.edgepedia.org/poincare-conjecture) was formulated using such invariants[12](https://maths.dur.ac.uk/users/dirk.schuetz/d46.pdf).\n\n## Later work: HOMFLYPT homology and categorified quantum groups\n\n**Khovanov–Rozansky homology.** With Lev Rozansky, Khovanov constructed, for each n ≥ 1, a bigraded link homology theory whose Euler characteristic is the quantum sl(n) polynomial, and the entire HOMFLY-PT polynomial is the Euler characteristic of a triply-graded link homology theory[4](https://ar5iv.labs.arxiv.org/html/math/0605339). The triply-graded construction assigns to a braid closure a complex of graded bimodules over a polynomial algebra in m generators, whose Hochschild homology is triply graded with Euler characteristic the HOMFLY-PT polynomial[4](https://ar5iv.labs.arxiv.org/html/math/0605339).\n\n**Categorified quantum groups.** With Joshua Sussan, Khovanov published \"A categorification of the positive half of quantum gl(m|1)\" in the Transactions of the AMS (2017)[13](https://www.math.columbia.edu/~khovanov/research/). The representation theory of Khovanov's arc algebras continues to develop: a November 2024 preprint proves that the Ext-quiver of the arc algebra H(m,n) has vertices labeled by the partitions λ in an (m×n)-rectangle that contain (m, m−1, …, 2, 1)[14](https://eprints.whiterose.ac.uk/id/eprint/244565/1/2411.15520v1.pdf).\n\n## By the numbers\n\nThe measurable influence of the construction is substantial. Computationally, the original construction computes Khovanov homology in a number of steps growing exponentially in the number of crossings; Dror Bar-Natan's tangle algorithms made it practically computable for links with 50 or more crossings[12](https://maths.dur.ac.uk/users/dirk.schuetz/d46.pdf). Because the Jones polynomial is easily recovered from Khovanov homology, computing Khovanov homology is at least as hard as computing the Jones polynomial, which is #P-hard for general link diagrams[12](https://maths.dur.ac.uk/users/dirk.schuetz/d46.pdf). There are exceptions: for every k, t ≥ 0 there is an algorithm computing the integral Khovanov homology of a t-strand braid closure in bounded homological degrees in polynomial time in the braid length, including a polynomial-time algorithm for 3-braid closures, while Bar-Natan's scanning algorithm runs in exponential time even on simple classes of 3-braids[12](https://maths.dur.ac.uk/users/dirk.schuetz/d46.pdf).\n\n## How it compares with other knot invariants\n\n**Against knot Floer homology.** Knot Floer homology, due to [Peter Ozsváth](https://www.edgepedia.org/peter-ozsvath), Zoltán Szabó, and Rasmussen, is constructed as a Lagrangian Floer homology in an auxiliary symplectic manifold; it categorifies the [Alexander polynomial](https://www.edgepedia.org/alexander-polynomial) and detects the unknot and the trefoils[7](https://ar5iv.labs.arxiv.org/html/1811.07848). Rasmussen conjectured that the rank of reduced Khovanov homology is at least the rank of reduced knot [Floer homology](https://www.edgepedia.org/floer-homology) for any knot in S³; this was proved via a spectral sequence from Khovanov homology to the delta-graded knot Floer homology of the mirror, over Q coefficients[7](https://ar5iv.labs.arxiv.org/html/1811.07848). The same theorem proves, up to one missing grading, the Dunfield–Gukov–Rasmussen conjecture on a spectral sequence from HOMFLY-PT homology to knot Floer homology[7](https://ar5iv.labs.arxiv.org/html/1811.07848). Khovanov lists a categorification of gl(1|1) as an open direction that should have deep relations to Heegaard Floer homology[11](http://media.scgp.stonybrook.edu/presentations/2023/20231107_Khovanov.pdf).\n\n**Against the Jones polynomial.** Khovanov homology contains strictly more information than the Jones polynomial, since there are pairs of knots with identical Jones polynomial but distinct Khovanov homology[6](https://google.iopscience.iop.org/article/10.1088/2632-072X/adde9f). E.-S. Lee proved that the ranks of the homology groups of alternating links are determined by the Jones polynomial and the signature, and programs computing Khovanov homology were written by Bar-Natan, A. Shumakovitch, and J. Green[4](https://ar5iv.labs.arxiv.org/html/math/0605339).\n\n## References\n\n1. [Mikhail Khovanov personal site.](https://sites.google.com/view/mkhovanov/)\n2. [Mikhail Khovanov, Institute for Advanced Study.](https://www.ias.edu/scholars/mikhail-khovanov)\n3. [Mikhail Khovanov. A categorification of the Jones polynomial. arXiv math.QA/9908171; Duke Math. J. 101 (2000), 359–426.](https://arxiv.org/abs/math.QA/9908171)\n4. [Mikhail Khovanov. Link homology and categorification (survey). arXiv math/0605339.](https://ar5iv.labs.arxiv.org/html/math/0605339)\n5. [Jozef H. Przytycki. Introduction to Khovanov homology. ICTS lecture notes, 2020.](https://www.icts.res.in/sites/default/files/Jozef%20H.%20Przytycki%20-%20ICTS%20-%20KT%20-%2026082020.pdf)\n6. [Khovanov Laplacian and Khovanov Dirac for knots and links. IOPscience, 2025.](https://google.iopscience.iop.org/article/10.1088/2632-072X/adde9f)\n7. [A Spectral Sequence from Khovanov Homology to Knot Floer Homology. arXiv 1811.07848.](https://ar5iv.labs.arxiv.org/html/1811.07848)\n8. [Mikhail Khovanov homepage, Columbia University Department of Mathematics.](https://math.columbia.edu/~khovanov/)\n9. [Mikhail Khovanov, Google Scholar profile.](https://scholar.google.com/citations?user=_PCGIMkAAAAJ&hl=en)\n10. [Categorical lifting of the Jones polynomial: a survey. AMS Bulletin 60 (2023), no. 4.](https://www.ams.org/journals/bull/2023-60-04/S0273-0979-2022-01772-7/viewer/)\n11. [Link homology and categorification. Mikhail Khovanov, Simons Center lecture, November 7, 2023.](http://media.scgp.stonybrook.edu/presentations/2023/20231107_Khovanov.pdf)\n12. [On computational complexity of Khovanov homology.](https://maths.dur.ac.uk/users/dirk.schuetz/d46.pdf)\n13. [Mikhail Khovanov research page, Columbia University.](https://www.math.columbia.edu/~khovanov/research/)\n14. [Quiver Presentations and Schur–Weyl Duality for Khovanov Arc Algebras. arXiv 2411.15520.](https://eprints.whiterose.ac.uk/id/eprint/244565/1/2411.15520v1.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Topologists and geometers › Low-dimensional and knot theorists*\n\n*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://math.columbia.edu/~khovanov/", "https://scholar.google.com/citations?user=_PCGIMkAAAAJ&hl=en", "https://www.math.columbia.edu/~khovanov/research/" ], "url": "https://www.edgepedia.org/mikhail-khovanov", "markdown_url": "https://www.edgepedia.org/mikhail-khovanov.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Mikhail Khovanov\", Edgepedia (EdgeChat), https://www.edgepedia.org/mikhail-khovanov. Edgepedia Community License 1.0.", "credit_md": "\"[Mikhail Khovanov](https://www.edgepedia.org/mikhail-khovanov)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/mikhail-khovanov](https://www.edgepedia.org/mikhail-khovanov). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Mikhail Khovanov\", Edgepedia (EdgeChat), https://www.edgepedia.org/mikhail-khovanov. Edgepedia Community License 1.0.", "speakable": "Mikhail Khovanov is a mathematician, formerly at Columbia and now a professor at Johns Hopkins University, who created Khovanov homology, a link invariant that categorifies the Jones polynomial and detects the unknot." }