Khovanov homology
Encyclopedia
In mathematics
, Khovanov homology is an invariant of oriented knots and links
that arises as the homology
of a chain complex
. It may be regarded as a categorification
of the Jones polynomial.
It was developed in the late 1990s by Mikhail Khovanov
, then at the University of California, Davis
, now at Columbia University
.
L, we assign the Khovanov bracket[ D] , a chain complex
of graded vector space
s. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise[ D] by a series of degree shifts (in the graded vector space
s) and height shifts (in the chain complex
) to obtain a new chain complex C(D). The homology
of this chain complex turns out to be an invariant
of L, and its graded Euler characteristic
is the Jones polynomial of L.
's paper.)
Let {l} denote the degree shift operation on graded vector spaces—that is, the homogeneous component in dimension m is shifted up to dimension m + l.
Similarly, let[ s] denote the height shift operation on chain complexes—that is, the rth vector space
or module
in the complex is shifted along to the (r + s)th place, with all the differential maps
being shifted accordingly.
Let V be a graded vector space with one generator q of degree 1, and one generator q−1 of degree −1.
Now take an arbitrary diagram D representing a link L. The axioms for the Khovanov bracket are as follows:
In the third of these, F denotes the `flattening' operation, where a single complex is formed from a double complex by taking direct sums along the diagonals. Also, D0 denotes the `0-smoothing' of a chosen crossing in D, and D1 denotes the `1-smoothing', analogously to the skein relation
for the Kauffman bracket.
Next, we construct the `normalised' complex C(D) =[ D] [ −n−] {n+ − 2n−}, where n− denotes the number of left-handed crossings in the chosen diagram for D, and n+ the number of right-handed crossings.
The Khovanov homology of L is then defined as the homology H(L) of this complex C(D). It turns out that the Khovanov homology is indeed an invariant of L, and does not depend on the choice of diagram. The graded Euler characteristic of H(L) turns out to be the Jones polynomial of L. However, H(L) has been shown to contain more information about L than the Jones polynomial, but the exact details are not yet fully understood.
In 2006 Dror Bar-Natan
developed a computer program to efficiently calculate the Khovanov homology (or category) for any knot.
of 3-manifolds. Moreover, it has been used to reprove a result only demonstrated using gauge theory
and its cousins: Jacob Rasmussen's new proof of a theorem of Kronheimer and Mrowka, formerly known as the Milnor conjecture
(see below). Conjecturally, there is a spectral sequence
relating Khovanov homology with the knot Floer homology
of Peter Ozsváth
and Zoltán Szabó
(Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover.
Khovanov homology is related to the representation theory of the Lie algebra
sl2. Mikhail Khovanov and Lev Rozansky have since defined cohomology
theories associated to sln for all n. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sln for all n.
Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology
, which they conjecture to be isomorphic to a singly-graded version of Khovanov homology. Ciprian Manolescu
has since simplified their construction and shown how to recover the Jones polynomial from the chain complex underlying his version of the Seidel-Smith invariant.
, and is sufficient to prove the Milnor conjecture.
In 2010, Kronheimer
and Mrowka
proved that the Khovanov homology detects the unknot
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Khovanov homology is an invariant of oriented knots and links
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
that arises as the homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
of a chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
. It may be regarded as a categorification
Categorification
In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional...
of the Jones polynomial.
It was developed in the late 1990s by Mikhail Khovanov
Mikhail Khovanov
Mikhail Khovanov is a professor of mathematics at Columbia University. He earned a PhD in mathematics from Yale University in 1997, where he studied under Igor Frenkel. His interests include knot theory and algebraic topology...
, then at the University of California, Davis
University of California, Davis
The University of California, Davis is a public teaching and research university established in 1905 and located in Davis, California, USA. Spanning over , the campus is the largest within the University of California system and third largest by enrollment...
, now at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
.
Overview
To any link diagram D representing a linkKnot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
L, we assign the Khovanov bracket
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
of graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
s. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
s) and height shifts (in the chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
) to obtain a new chain complex C(D). The homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
of this chain complex turns out to be an invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
of L, and its graded Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
is the Jones polynomial of L.
Definition
(This definition follows the formalism given in Dror Bar-NatanDror Bar-Natan
Dror Bar-Natan is a mathematics professor at the University of Toronto, Canada. His main research interests include knot theory, finite type invariants, and Khovanov homology.-Education:...
's paper.)
Let {l} denote the degree shift operation on graded vector spaces—that is, the homogeneous component in dimension m is shifted up to dimension m + l.
Similarly, let
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
or module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
in the complex is shifted along to the (r + s)th place, with all the differential maps
Differential (mathematics)
In mathematics, the term differential has several meanings.-Basic notions:* In calculus, the differential represents a change in the linearization of a function....
being shifted accordingly.
Let V be a graded vector space with one generator q of degree 1, and one generator q−1 of degree −1.
Now take an arbitrary diagram D representing a link L. The axioms for the Khovanov bracket are as follows:
-
[ ø] = 0 → Z → 0, where ø denotes the empty link. -
[ O D] = V ⊗[ D] , where O denotes an unlinked trivial component. -
[ D] = F(0 →[ D0] →[ D1] {1} → 0)
In the third of these, F denotes the `flattening' operation, where a single complex is formed from a double complex by taking direct sums along the diagonals. Also, D0 denotes the `0-smoothing' of a chosen crossing in D, and D1 denotes the `1-smoothing', analogously to the skein relation
Skein relation
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial which is an invariant of the knot. If two diagrams have different polynomials, they represent different knots. The reverse may not...
for the Kauffman bracket.
Next, we construct the `normalised' complex C(D) =
The Khovanov homology of L is then defined as the homology H(L) of this complex C(D). It turns out that the Khovanov homology is indeed an invariant of L, and does not depend on the choice of diagram. The graded Euler characteristic of H(L) turns out to be the Jones polynomial of L. However, H(L) has been shown to contain more information about L than the Jones polynomial, but the exact details are not yet fully understood.
In 2006 Dror Bar-Natan
Dror Bar-Natan
Dror Bar-Natan is a mathematics professor at the University of Toronto, Canada. His main research interests include knot theory, finite type invariants, and Khovanov homology.-Education:...
developed a computer program to efficiently calculate the Khovanov homology (or category) for any knot.
Related theories
One of the most interesting aspects of Khovanov's homology is that its exact sequences are formally similar to those arising in the Floer homologyFloer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
of 3-manifolds. Moreover, it has been used to reprove a result only demonstrated using gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
and its cousins: Jacob Rasmussen's new proof of a theorem of Kronheimer and Mrowka, formerly known as the Milnor conjecture
Milnor conjecture (topology)
In knot theory, the Milnor conjecture says that the slice genus of the torus knot is/2.It is in a similar vein to the Thom conjecture....
(see below). Conjecturally, there is a spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
relating Khovanov homology with the knot Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
of Peter Ozsváth
Peter Ozsváth
Peter Steven Ozsváth is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds....
and Zoltán Szabó
Zoltán Szabó
Zoltán Szabó is a professor of mathematics at Princeton University. He created, along with Peter Ozsváth, Heegaard Floer homology, a homology theory for 3-manifolds. For this contribution to the field of topology, Ozsváth and Szabó were awarded the 2007 Oswald Veblen Prize in Geometry.He got his...
(Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover.
Khovanov homology is related to the representation theory of the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
sl2. Mikhail Khovanov and Lev Rozansky have since defined cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
theories associated to sln for all n. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sln for all n.
Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
, which they conjecture to be isomorphic to a singly-graded version of Khovanov homology. Ciprian Manolescu
Ciprian Manolescu
Ciprian Manolescu is a Romanian mathematician. He is presently an Associate Professor in the mathematics department at the University of California, Los Angeles....
has since simplified their construction and shown how to recover the Jones polynomial from the chain complex underlying his version of the Seidel-Smith invariant.
Applications
The first application of Khovanov homology was provided by Jacob Rasmussen, who defined the s-invariant using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genusSlice genus
In mathematics, the slice genus of a smooth knot K in S3 is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g embedded in the 4-ball D4 bounded by S3.More precisely, if S is required to be smoothly embedded, then this integer g is the...
, and is sufficient to prove the Milnor conjecture.
In 2010, Kronheimer
Peter B. Kronheimer
Peter Benedict Kronheimer is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is presently William Casper Graustein Professor of Mathematics at Harvard University....
and Mrowka
Tomasz Mrowka
Tomasz Mrowka is a Polish American mathematician. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology since 2010. A graduate of MIT, he received the Ph.D. from University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby...
proved that the Khovanov homology detects the unknot
Unknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
.