Categorical lifting of the Jones polynomial: a survey
This paper is dedicated to the memory of Vaughan Jones,whose insights have illuminated so many beautiful mathematical paths.
Abstract
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.
1. Constructions of the Jones polynomial
The spectacular discovery by Vaughan Jones Reference 76Reference 78 of the Jones polynomial of links has led to many follow-up developments in mathematics. In this note we will survey one of these developments, the discovery of a combinatorially defined homology theory of links, functorial under link cobordisms in 4-space, and its connections to algebraic geometry, symplectic geometry, gauge theory, representation theory, and stable homotopy theory.
The Jones polynomial of an oriented link in is determined uniquely by the skein relation
and the normalization that the polynomial of the unknot satisfies . The multiplicativity property (that is, that the disjoint union with the unknot scales the invariant by ) suggests another natural normalization, and , where is the empty link.
The polynomial originally arose from Jones’s work on -algebras, where the braid relations and Temperley–Lieb relations appeared organically Reference 75Reference 77. As we will see below, it also has connections to many other areas, from representation theory to gauge theory. Many of these connections first appeared or were foreshadowed in papers of Jones’s, including the connections to quantum groups and statistical mechanics Reference 79, Hecke algebras and traces Reference 77Reference 78, and many other topics Reference 80. In addition to inspiring at least half a dozen different fields in mathematics, the Jones polynomial and its descendants have had remarkable applications to topology. Some we will touch on below; others, like its central role in resolving the famous Tait conjectures or its deep connections to hyperbolic geometry, we leave to other authors.
While it is fairly easy to see that at most one knot invariant satisfies relation Equation 1 and any given normalization for , it is not immediately obvious that Equation 1 is consistent. A simple way to see the existence of a knot invariant satisfying Equation 1 was discovered by L. Kauffman Reference 81. Pick a planar diagram of , forget about the orientation of , and resolve each crossing of into a linear combination of two crossingless diagrams, as shown in Figure 1.1. Any time a simple closed curve without crossings arises, remove it and scale the remaining term by . The end result is a Laurent polynomial (where is the number of crossings of ), the Kauffman bracket of . We can now bring back the orientation of and scale by a monomial in terms of the number of positive crossings and of negative crossings (the first and second pictures in formula Equation 1):
It is straightforward to check that is invariant under Reidemeister moves of oriented link diagrams, hence it gives rise to a link invariant . Further, by applying the unoriented skein relation from Figure 1.1 at the crossing of the two diagrams on the left of relation Equation 1, one sees that satisfies relation Equation 1. So, we have:
For any oriented link , .
2. Categorification of the Jones polynomial for links and tangles
2.1. Categorification for links
E. Witten showed Reference 176 at a physical level of rigor that the Chern–Simons path integral, with gauge group and parameter a root of unity, gives rise to an invariant of 3-manifolds intricately related to the Jones polynomial. The case of gauge group was considered earlier by A. Schwarz, who showed that the path integral evaluates to the Reidemeister torsion Reference 157. Shortly afterward, N. Reshetikhin and V. Turaev Reference 147 gave a mathematically precise proof that suitable linear combinations of the Jones polynomial of cables of a framed link , evaluated at an th root of unity, give invariants of an oriented 3-manifold obtained by surgery on ; the resulting invariants are called Witten–Reshetikhin–Turaev invariants.
Motivated by these developments and by constructions in geometric representation theory (notably by the work of G. Lusztig Reference 114 and A. Beilinson, Lusztig, and R. MacPherson Reference 20), L. Crane and I. B. Frenkel conjectured Reference 38 that the Witten–Reshetikhin–Turaev -manifold invariant lifts to a four-dimensional topological quantum field theory (TQFT). They coined the term categorification to describe such a lifting of an -dimensional TQFT to an -dimensional TQFT.
Despite many insights into the possible structure of such a theory since then, its existence still remains a conjecture. Nonetheless, the Crane–Frenkel conjecture motivated the discovery of a categorification of the Jones polynomial by the first author Reference 85. In that categorification, the parameter becomes a grading shift of the quantum grading, and the theory assigns to an oriented link bigraded homology groups
functorial under smooth link cobordisms, and with the Jones polynomial as their Euler characteristic:
A way to construct this theory can be guessed by lifting the Kauffman skein relation to a long exact sequence for homology. That is, up to appropriate grading shifts, there is an exact sequence
Suppose further that, given a diagram for , there is a chain complex computing , and the long exact sequence is induced by an isomorphism between the complex and the cone of a map between and , where and are as in Figure 1.1. The Jones invariant of the unknot is , which is the graded rank of a free graded abelian group with generators in degrees and . The philosophy of TQFTs then suggests associating to a -component unlink diagram. Natural maps between these complexes for resolutions of can be obtained from a commutative Frobenius algebra structure on : change of resolution is a cobordism, and Frobenius algebras correspond to two-dimensional TQFTs, assigning maps to cobordisms between -manifolds. It turns out that is unique up to obvious symmetries: with generators in (quantum) degrees and denoted by and , respectively, the multiplication and the trace on are given by
Dualizing the multiplication via leads to a comultiplication, with
Explicitly, and allow one to write down maps associated to all local topology changes between full resolutions of an -crossing diagram , giving a commutative -dimensional cube with powers of at its vertices and maps and tensored with identity maps on its edges. After suitable degree shifts, by collapsing the cube (similar to passing to the total complex of a polycomplex) one obtains a complex of graded abelian groups with a differential that preserves the quantum degree. Reidemeister moves can be lifted to specific homotopy equivalences between the complexes. Consequently, the isomorphism class of the bigraded homology groups is an invariant of , now widely called homology or Khovanov homology. Identification of the Jones polynomial as the Euler characteristic of is immediate, since the construction of lifts Kauffman’s inductive formula.
One can think of this construction of a link homology as coming from a commutative Frobenius algebra over , as above. The key property of is having rank over the ground ring : using an algebra of larger rank, the homology fails to be invariant under Reidemeister I moves. On the other hand, a modification of this construction, deforming the relation , gives rise to so-called equivariant link homology Reference 16Reference 90. The essentially most general deformation comes from working over the ground ring and setting to be
The equivariant theory turns out to be important for applications (see Sections 3 and 5).
As mentioned above, this construction of link homology can be phrased via a rank commutative Frobenius pair , giving rise to a two-dimensional TQFT with (with is or above) and . That a two-dimensional TQFT of rank can be bootstrapped into a link homology theory was surprising.
There is also a reduced version of the invariant, corresponding to the normalization . Fix a marked point on a strand of . There is a subcomplex where the marked circle is labeled throughout. Shifting the quantum grading of down by and taking homology gives , the reduced Khovanov homology. It is easy to see that , so there is a long exact sequence
A paper of D. Bar-Natan Reference 15 helped to provoke early interest in the subject, as well as giving computations of for knots through 12 crossings. (More work on computing is described in Section 5.)
2.2. Tangles and representations
The Kauffman bracket invariant admits a relative version for tangles in the 3-disk Reference 32Reference 81Reference 82Reference 83. Start with a tangle in with boundary points, and consider a generic projection of it to the 2-disk , with boundary points spread out around the boundary . Let be the free -module with basis the set of crossingless matchings of boundary points via disjoint arcs inside a disk. The relative Kauffman bracket associates to an element of by resolving each crossing following Kauffman’s recipe. The braid group on strands acts on by attaching a braid to a crossingless matching and then reducing the result via Kauffman’s relations. (In fact, the larger group of annular braids acts.) More generally, a tangle in a strip with bottom and top points (a -tangle) induces a -linear map
These maps fit together into a functor from the category of even tangles (tangles with an even number of top and bottom endpoints) to the category of -modules. Variations of Kauffman’s construction can be made into monoidal functors from the category of tangles that assign th tensor power of the fundamental representation of quantum (or a suitable subspace of ) to points on the plane and intertwiners between tensor powers of representations to tangles. The above setup with crossingless matchings corresponds to assigning the subspace of invariants to points. This subspace is trivial when is odd and has a basis of crossingless matchings for even Reference 32Reference 53Reference 83Reference 84.
Upon categorification, becomes a Grothendieck group of a suitable category . A crossingless matching with specified endpoints becomes an object of . We can guess that morphism spaces will come from cobordisms between and , that is, surfaces embedded in with boundaries , , and . (An example is on the right of Figure 2.1.) The total boundary of such a surface is homeomorphic to the -manifold given by gluing and along their boundary points. One can then define
by applying the two-dimensional TQFT as above to that -manifold. It is straightforward to define associative multiplications
by applying to appropriate cobordisms Reference 86.
More carefully, to define we start with objects and morphisms as above and form a pre-additive category . Equivalently, category can be viewed as an idempotented ring
the arc ring, with idempotents given by identity cobordisms from to itself. It is also possible to keep track of morphisms in different degrees and refine the category by restricting morphisms to degree parts of graded abelian groups but allowing grading shifts of generating objects to capture the entire groups.
From the idempotents one can recover the projective modules over .
One can then form an additive closure of the category by also allowing finite direct sums of objects. The category happens to be Karoubi closed, which is not hard to check and simplifies working with it. The category is equivalent to the category of graded projective finitely generated modules over the graded ring .
To a flat (crossingless) tangle in a disk with endpoints there is associated an object of or, equivalently, a projective graded -module. If is the union of circles and a crossingless matching , then the projective module is isomorphic to , that is, to the sum of copies of the projective module , with appropriate grading shifts.
The Grothendieck group of is a free -module with basis given by the symbols of projective modules, over all crossingless matchings . This Grothendieck group can also be defined as of the graded algebra . There is a canonical isomorphism of -modules
Now form the category of bounded complexes of objects of , modulo chain homotopies. The inclusion induces an isomorphism of their Grothendieck groups.
To a planar diagram of a tangle with endpoints there is an associated object of , by a relative version of the cube construction. Namely, define to be the iterated mapping cone of the two resolutions at each crossing, that is, the total complex of the cube of resolutions of . See Figure 2.1 for a simple example.
Reidemeister moves of tangle diagrams lift to chain homotopy equivalences, and the isomorphism class of the object is an invariant of . On the Grothendieck group, descends to the element .
Similarly, given a tangle diagram with bottom and top endpoints, there is an associated complex of )-bimodules, and tensoring with this complex of bimodules gives an exact functor . This construction lifts to a 2-functor from the category of flat tangles and their cobordisms to the category of bimodules and their homomorphisms. Furthermore, it lifts to a projective functor (well-defined on 2-morphisms up to an overall sign) from the 2-category of tangle cobordisms to the 2-category of complexes of bimodules over , over all , and maps of complexes, up to homotopy Reference 16Reference 89 (see also Reference 72 for another proof). Taking care of the sign is subtle; see Reference 23Reference 31Reference 35Reference 152.
Categories of representations of the arc rings categorify
It turns out that the entire tensor product , as well as the commuting actions of the Temperley–Lieb algebra and quantum on it, can also be categorified. This categorification was realized in Reference 22 via maximal singular and parabolic blocks of highest weight categories for , with the commuting actions lifting to those by projective functors and Bernstein–Zuckerman functors (see also Reference 52Reference 185).
The tensor power decomposes as the sum of its weight spaces , . A more explicit categorification of weight spaces and the Temperley–Lieb algebra action on them can be achieved via specific subquotient rings of Reference 27Reference 34. J. Brundan and C. Stroppel showed Reference 28Reference 29 that these subquotient rings
- (a)
describe maximal parabolic blocks of highest weight categories for , relating the two categorifications, and
- (b)
describe blocks of representations of Lie superalgebras .
The space of invariants is naturally a subspace of the middle weight space . Analogues of this subspace for a general weight space are given by the kernel of the generator for , and the kernel of for . Categorifications of these subspaces are provided by representation categories of certain Frobenius algebras, like , that can be obtained as subquotients of . The latter Frobenius algebras as well as Morita and derived Morita equivalent algebras are widespread in modular representation theory. For instance, Hiss and Lux’s book Reference 68 lists hundreds of examples of blocks of finite groups over finite characteristic fields that are (derived) Morita equivalent to the self-dual part of the zigzag algebra from Reference 91, the latter giving a categorification of the reduced Burau representation of the braid group and of the corresponding subspace of the first nontrivial weight space, .
A very general framework for a categorification of tensor products of quantum group representations and Reshetikhin–Turaev link invariants was developed by Ben Webster Reference 173. The case of his construction Reference 172 uses algebras that are Morita equivalent to Koszul duals of the above-mentioned subquotients of .
2.3. Connections to algebraic geometry, symplectic geometry, and beyond
The connection with representation theory inspired a further connection with symplectic geometry. Given a symplectic manifold , there is an associated triangulated category, the derived Fukaya category. The objects of the Fukaya category are Lagrangian submanifolds of (with certain extra data), and the morphism spaces are categorified intersection numbers, defined via Floer theory. Given a braid group action on , there is an induced braid group action on the Fukaya category and, hence, potentially, a knot invariant. The first examples of such braid group actions were given by P. Seidel and the first author Reference 91. Soon after, Seidel and I. Smith gave a braid group action on a more complicated, but natural, symplectic manifold, and from it a conjectured Floer-theoretic definition of Khovanov homology, which they called symplectic Khovanov homology Reference 160. (See also Reference 116 for a reinterpretation of this construction.) Recently, M. Abouzaid and Smith proved that this conjecture holds over Reference 1Reference 2. The proof uses the extension of Khovanov homology to tangles discussed above to identify the two theories. At present, it is unknown whether the torsion in symplectic Khovanov homology and in combinatorial Khovanov homology agree. Although it is harder to compute, symplectic Khovanov homology is in some ways more geometric. In particular, its relationship to Heegaard Floer homology and its behavior for periodic knots (see Section 3), as well as the equivariant versions of the theory in the sense of Equation 8, all have geometric definitions via group actions on the symplectic manifold Reference 67Reference 161.
The symplectic manifolds in the Seidel–Smith construction are examples of quiver varieties, so carry hyperkähler structures. Complex Lagrangians determine objects of both the Fukaya category and the category of coherent sheaves with respect to the rotated almost-complex structure. The fact that automorphism algebras on the two sides are isomorphic to ordinary cohomology can be seen as a shadow of mirror symmetry and can often be lifted to an equivalence of categories. Consequently, one would expect that the tangle extension of Khovanov homology can be realized via derived categories of coherent sheaves on the corresponding quiver varieties, with functors associated to tangles acting via suitable Fourier–Mukai kernels (convolutions with objects of the derived category on the direct product of varieties). A modification of this idea was realized by S. Cautis and J. Kamnitzer Reference 33. They use certain smooth completions of these quiver varieties which can be realized as iterated -bundles and interpreted as convolution varieties of the affine Grassmannian for , also providing a connection to the geometric Satake correspondence. The relation to quiver varieties and the -Springer fiber has been established by R. Anno Reference 7 and by Anno and V. Nandakumar Reference 8, who also explained the relation between coherent sheaves on these varieties and the rings and their annular versions. An isomorphism between the center of and the cohomology ring of the -Springer fiber, established in Reference 88, was an earlier indication of the connection between the two structures.
There has been strong interest in giving physical reinterpretations and extensions of link homology invariants. One program to do so was initiated by Witten, using the Kapustin–Witten and Haydys–Witten equations Reference 177. Other proposals have been put forward by S. Gukov, A. Schwarz, and C. Vafa Reference 63; Gukov, P. Putrov, and Vafa Reference 62; Gukov, D. Pei, Putrov, and Vafa Reference 61; M. Aganagic Reference 3Reference 4; and others.
Currently, Khovanov homology is only defined for links in a few manifolds: , as described above; links in thickened surfaces, in work of Asaeda, Przytycki, and Sikora Reference 9; and links in connected sums of , in work of Rozansky Reference 150 and Willis Reference 175. (See also the universal construction in Reference 124.) One appeal of some of the conjectural physical approaches to Khovanov homology is that they may apply in general 3-manifolds. In a recent paper Reference 144, J. Sussan and Y. Qi categorify the Jones polynomial when the quantum parameter is a prime root of unity; this is also related to extending Khovanov homology to other 3-manifolds.
There is a large literature on categorification of representations and quantum invariants, for an arbitrary . For lack of space, we will not discuss these developments in this paper. Nor do we discuss the related topics of annular homology, categorifications of the colored Jones polynomial, foams, and categorified quantum groups.
3. Signs and spectral sequences
One reason Khovanov homology has been important is that it seems to be a kind of free object in the category of knot homologies, a property which is witnessed by the many spectral sequences from Khovanov homology to other knot homologies. (An attempt to make precise the sense in which Khovanov homology is free was given in Reference 11.) These spectral sequences often connect invariants whose constructions appear quite different, in some cases giving relationships between invariants that are not apparent at the classical, decategorified level. They have led to many of the topological applications of Khovanov homology, as well as to new properties of Khovanov homology itself.
The first spectral sequence from Khovanov homology was constructed by E. S. Lee Reference 101 (see also Reference 146). Recall the family of deformations of the Frobenius algebra from equation Equation 8. Taking the parameters and extending scalars from to , we obtain the algebra . The quantum grading weakens to a filtration on the resulting complex, inducing a spectral sequence from Khovanov homology to this deformed knot invariant, called Lee homology. To understand Lee homology, note that this Frobenius algebra diagonalizes, as a direct sum of two one-dimensional Frobenius algebras. It follows easily that the Lee homology of a -component link has dimension . Using this construction, Lee verified a conjecture of Bar-Natan Reference 15, S. Garoufalidis Reference 54, and the first author Reference 87 that the Khovanov homology of an alternating knot lies on two adjacent diagonals. More famous applications of this spectral sequence are discussed in Section 5.
Most of the other spectral sequences from Khovanov homology relate to gauge theory. The first of these is due to P. Ozsváth and Z. Szabó Reference 137. Given a closed, oriented -manifold , they had constructed an abelian group , the homology of a chain complex Reference 136. Inspired by A. Floer’s exact triangle Reference 48, they showed that given a knot and slopes , , and on intersecting each other pairwise once, there is an exact triangle relating the Floer homologies of the surgeries , , and Reference 135. In particular, given a link in , if and are the - and -resolutions of a crossing of , then the surgery exact triangle gives an exact triangle of Floer homologies of their branched double covers,
(This is an ungraded exact triangle: the groups do not have canonical -gradings, and the gradings they do have are not respected by the maps in the exact triangle.) The surgery exact triangle is local, in the sense that given disjoint links and , the maps in the surgery exact triangles associated to and commute or, at the chain level, commute up to reasonably canonical homotopy. So, resolving all crossings of gives a cube of resolutions for . The -page of the associated spectral sequence is
which has the same dimension as the reduced Khovanov complex. The differential on the -page comes from merge and split cobordisms
These maps correspond to some two-dimensional Frobenius algebra which, in fact, turns out to be the algebra . Thus, one obtains a spectral sequence from the reduced Khovanov homology of (the mirror of) , with -coefficients, to .
The Euler characteristic of is the number of elements in if finite, or otherwise. So, the Ozsváth–Szabó spectral sequence lifts the equality .
To summarize, the key properties of used to construct the Ozsváth–Szabó spectral sequence were the existence of an unoriented skein triangle satisfying a far-commutativity property; TQFT properties for disjoint unions, merges, and splits; and the fact that its value on an unknot (or, more accurately, 2-component unlink) is a two-dimensional vector space.
In 2010, P. Kronheimer and T. Mrowka built a gauge-theoretic invariant with these properties, using Donaldson theory Reference 96. Like many gauge-theoretic invariants, the value of constrains how surfaces can be embedded. Using this, Kronheimer and Mrowka deduced that if the genus of a knot is , then has dimension . From the argument above, there is a spectral sequence , hence:
If , then is the unknot.
The stronger, and older, conjecture, that only if is the unknot, remains open.
There are many other spectral sequences from Khovanov homology, including more variants of the Lee spectral sequence Reference 14Reference 42, spectral sequences defined using instanton and monopole Floer homology Reference 25Reference 39Reference 155, other spectral sequences defined via variants of Heegaard Floer homology Reference 58Reference 148, spectral sequences coming from equivariant symplectic Khovanov homology and equivariant Khovanov homology Reference 36Reference 161Reference 167Reference 184, and a combinatorial spectral sequence conjectured to agree with the Ozsváth–Szabó spectral sequence Reference 169 (see also Reference 154). This last spectral sequence also supports another conjecture: that the Ozsváth–Szabó spectral sequence preserves the -grading on Khovanov homology Reference 56. Another notable spectral sequence is due to J. Batson and C. Seed Reference 19: given a link , they construct a spectral sequence to the disjoint union of the sublinks and (which is just if working over a field). The page of collapse of this spectral sequence gives a lower bound on the unlinking number of . It and many of the other spectral sequences have also been used to prove further detection results for Khovanov homology, in the spirit of Theorem 3.1. Often, the proofs of detection results combine several of these spectral sequences. Some examples of such results include:
Let be the -component unlink. If
for all and , then is isotopic to .
The proof uses Theorem 3.1 and the Batson–Seed spectral sequence. A related result was obtained earlier by M. Hedden and Y. Ni Reference 65. (By contrast, the Jones polynomial does not detect the unlink Reference 43Reference 171. Indeed, most of the detection results mentioned below also do not hold for the Jones polynomial.)
If is an -component link with , then is a forest of Hopf links.
The proof uses Kronheimer and Mrowka’s spectral sequence and its extension to annular links Reference 179 (building on Reference 9Reference 58Reference 149); Batson and Seed’s spectral sequence; and N. Dowlin’s spectral sequence mentioned below. In other papers, the authors classify all links with Khovanov homology of dimension Reference 182 and show that Khovanov homology detects, for instance, Reference 105. Similarly, Khovanov homology detects the link Reference 120.
Let be a knot.
- (1)
If , then