ScalingStacks

During the past 15 years many new structures have arisen in the topology of low-dimensional manifolds: the Jones and HOMFLY polynomials of links, Witten-Reshetikhin-Turaev invariants of 3-manifolds, Floer homology groups of homology 3-spheres, Donaldson and Seiberg-Witten invariants of 4-manifolds. These invariants of 3- and 4-manifolds naturally split into two groups. Members of the first group are combinatorially defined invariants of knots and 3-manifolds, such as various link polynomials, finite type invariants and quantum invariants of 3-manifolds. Floer and Seiberg-Witten homology groups of 3-manifolds and Donaldson-Seiberg-Witten invariants of 4-manifolds constitute the second group. While invariants from the first group have a combinatorial description and in each instance can be computed algorithmically, invariants from the second group are understood through moduli spaces of solutions of suitable differential-geometric equations and the infinite-dimensional Morse theory and have evaded all attempts at a finite combinatorial definition. These invariants have been computed for many 3- and 4-manifolds, yet the methods of computation use some extra structure on these manifolds, such as Seifert fibering or complex structure and the problem of finding an algorithmic construction of these invariants remains open.

It is probably due to this striking difference in the origins and computational complexity that so far not many direct relations have been found between invariants from different groups. The most notable connection is the Casson invariant of homology 3-spheres [AM], which is equal to the Euler characteristic of Floer homology [F]. Yet the Casson invariant is computable and intimately related to the Alexander polynomial of knots and (see [M]) to Witten-Reshetikhin-Turaev invariants, which are examples of invariants from the first group. A similar relation has recently been discovered between Seiberg-Witten invariants and Milnor torsion of 3-manifolds ([MT]). In summary, Euler characteristics of Floer and Seiberg-Witten homology groups bear an algorithmic description, while no such procedure is known for finding the groups themselves.

A speculative question now comes to mind: quantum invariants of knots and 3-manifolds tend to have good integrality properties. What if these invariants can be interpreted as Euler characteristics of some homology theories of 3-manifolds?

Our results suggest that such an interpretation exists for the Jones polynomial of links in 3-space ([Jo]). We give an algorithmic procedure that to a generic plane projection $D$ of an oriented link $L$ in $\mathbb{R}^{3}$ associates cohomology groups ${\cal H}^{i,j}(D)$ that depend on two integers $i,j.$ If two diagrams $D_{1}$ and $D_{2}$ of the same link $L$ are related by a Reidemeister move, a canonical isomorphism of groups ${\cal H}^{i,j}(D_{1})$ and ${\cal H}^{i,j}(D_{2})$ is constructed. Thus, isomorphism classes of these groups are invariants of the link $L.$ These groups are finitely generated and may have non-trivial torsion. Tensoring these groups with $\mathbb{Q}$ we get a two-parameter family $\{\mbox{dim}_{\mathbb{Q}}({\cal H}^{i,j}(D)\otimes\mathbb{Q})\}_{i,j\in\mathbb{Z}}$ of integer-valued link invariants.

From our construction of groups ${\cal H}^{i,j}(D)$ we immediately conclude that the graded Euler characteristic

is equal, up to a simple change of variables, to the Jones polynomial of $L,$ multiplied by $q+q^{-1}.$

We conjecture that not just the isomorphism classes of ${\cal H}^{i,j}(D)$ but the groups themselves are invariants of links. We will consider this conjecture in a subsequent paper.

To define cohomology groups ${\cal H}^{i,j}(D)$ we start with the Kauffman state sum model [Ka] for the Jones polynomial and then, roughly speaking, turn all integers into complexes of abelian groups. In the Kauffman model a link is projected generically onto the plane so that the projection has a finite number of double transversal intersections. There are two ways to “smooth” the projection near the double point, i.e., erase the intersection of the projection with a small neighbourhood of the double point and connect the four resulting ends by a pair of simple, nonintersecting arcs:

A diagram $D$ with $n$ double points admits $2^{n}$ resolutions of these double points. Each of the resulting diagrams is a collection of disjoint simple closed curves on the plane. In [Ka] Kauffman associates Laurent polynomial $(-q-q^{-1})^{k}$ to a collection of $k$ simple curves and then forms a weighted sum of these numbers over all $2^{n}$ resolutions. After normalization, Kauffman obtains the Jones polynomial of the link $L.$ The principal constant in this construction is $-q-q^{-1},$ the number associated to a simple closed curve.

In our approach $q+q^{-1}$ becomes a certain module $A$ over the base ring $\mathbb{Z}[c].$ In detail, we work over the graded ring $\mathbb{Z}[c]$ of polynomials in $c,$ where $c$ has degree $2,$ and we define $A$ to be a free $\mathbb{Z}[c]$-module of rank two with generators in degrees $1$ and $-1.$ This is the object we associate to a simple closed curve in the plane.

Given a diagram $D$, to each resolution of all double points of $D$ we associate the graded $\mathbb{Z}[c]$-module $A^{\otimes k},$ where $k$ is the number of curves in the resolution. Then we glue these modules over all $2^{n}$ resolutions into a complex $C(D)$ of graded $\mathbb{Z}[c]$-modules. The gluing maps come from commutative algebra and cocommutative coalgebra structure on $A.$ When two diagrams $D_{1}$ and $D_{2}$ are related by a Reidemeister move, we construct a quasi-isomorphism between the complexes $C(D_{1})$ and $C(D_{2}).$

The cohomology groups $H^{i}(D)$ of the complex $C(D)$ are graded $R$-modules and we prove that isomorphism classes of $H^{i}(D)$ do not depend of the choice of a diagram of the link. We then look at some elementary properties of these groups and introduce several cousins of $H^{i}(D).$

Outline of the paper. In Section 2 we define an algebra $A$ over the ring $R=\mathbb{Z}[c]$ and use $A$ to construct a 2-dimensional topological quantum field theory. In our case this topological quantum field theory is a functor from the category of two-dimensional cobordisms between one-dimensional manifolds to the category of graded $\mathbb{Z}[c]$-modules. In Section 2.4 we review the Kauffman state sum model [Ka] for the Jones polynomial of oriented links.

In Section 4.2, we associate a complex of $\mathbb{Z}[c]$-modules to a plane diagram of a link. As an intermediate step, to a diagram $D$ we associate a commutative cube $V_{D}$ of $\mathbb{Z}[c]$-modules and maps between them, i.e., we consider an $n$-dimensional cube with its edges standardly oriented and, given a plane projection with $n$ double points of a link, to each vertex of the cube we associate a $\mathbb{Z}[c]$-module and to each oriented edge a map of modules so that all square facets of this diagram are commutative squares. This is done in Section 4.2. In the same section we pass from commutative cubes to complexes of $\mathbb{Z}[c]$-modules and to a diagram $D$ we associate a complex $C(D)$ of graded $\mathbb{Z}[c]$-modules.

Earlier, in Section 3, we review the notions of a commutative cube and a map between commutative cubes.

In Section 4.1 we review Reidemeister moves. In Section 5, which is the technical core of the paper, to a Reidemeister move between diagrams $D_{1}$ and $D_{2}$ we associate a quasi-isomorphism between the complexes $C(D_{1})$ and $C(D_{2}).$ These isomorphisms seem to be canonical. We conjecture that the quasi-isomorphisms are coherent, which would naturally associate cohomology groups to links. Our quasi-isomorphism result shows that the isomorphism classes of the cohomology groups are invariants, but not necessarily that the groups are functorial under link isotopy.

We define $H^{i}(D)$ to the be $i$-th cohomology group of the complex $C(D).$ These cohomology groups are graded $\mathbb{Z}[c]$-modules, and the isomorphism class of each $H^{i}(D)$ is a link invariant. If we split these groups into the direct sum of their graded components,

we get a two-parameter family of “abelian group valued” link invariants. These results are stated earlier, at the end of Section 4.2, as Theorems 1 and 2. For a diagram $D,$ the groups $H^{i,j}(D)$ are trivial for $j\ll 0.$ Moreover, for each $j$ only finitely many of the groups $H^{i,j}(D)$ are non-zero. Consequently, the graded Euler characteristic of $C(D),$ defined as

is well-defined as a Laurent series in $q.$ Since our construction of $C(D)$ lifts Kauffman’s construction of the Jones polynomial, it is not surprising that the graded Euler characteric of $C(D)$ is related to the Jones polynomial. Namely, $\widehat{\chi}(C(D)),$ multiplied by $\frac{1-q^{2}}{q+q^{-1}},$ is equal, after a simple change of variables, to the Jones polynomial of $L.$

In Section 7 we explain how a version of our construction when the base algebra $\mathbb{Z}[c]$ is reduced to $\mathbb{Z}$ by taking $c=0$ produces graded cohomology groups ${\cal H}^{i,j}(D).$ The complex which is used to define ${\cal H}^{i,j}(D)$ is given by tensoring $C(D)$ with $\mathbb{Z}$ over $\mathbb{Z}[c].$ As before, the isomorphism classes of these groups are invariants of links. These groups are “smaller” than the groups $H^{i,j}(D).$ In particular, for each $D,$ these groups are non-zero for only finitely many pairs $(i,j)$ of integers. As with the groups groups $H^{i,j}(D),$ the graded Euler characteristic

divided by $q+q^{-1}$, is equal to the Jones polynomial of the link represented by the diagram $D.$ In Section 7.5 we exhibit a spectral sequence whose $E_{1}$-term is made of ${\cal H}(D)$ and which converges to $H(D).$ Apparently, $H(D)$ is a kind of $S^{1}$-equivariant version of the groups ${\cal H}^{i,j}(D).$ In Section 7.7 we use these cohomology groups to reprove a result of Thistlethwaite on the crossing number of adequate links.

If a link in $\mathbb{R}^{3}$ has cohomology groups, then cobordisms between links, i.e., surfaces embedded in $\mathbb{R}^{3}\times[0,1],$ should provide maps between the associated groups. A surface embedded in the 4-space can be visualized as a sequence of plane projections of its 3-dimensional sections (see [CS]). Given such a presentation $J$ of a compact oriented surface $S$ properly embedded in $\mathbb{R}^{3}\times[0,1]$ with the boundary of $S$ being the union of two links $L_{0}\subset\mathbb{R}^{3}\times\{0\}$ and $L_{1}\subset\mathbb{R}^{3}\times\{1\},$ we explain in Section 6.3 how to associate to $J$ a map of cohomology groups

$\chi(S)$ being the Euler characteristic of the surface $S$ and $D_{0}$ and $D_{1}$ – diagrams of $L_{0}$ and $L_{1}$ induced by $J.$ We conjecture that, up to an overall minus sign, this map does not depend on the choice of $J,$ in other words, $\pm\theta_{J}$ behaves invariantly under isotopies of $S.$

If this conjecture is true, we get a 4-dimensional topological quantum field theory, restricted to links in $\mathbb{R}^{3}$ and $\mathbb{R}^{3}\times[0,1]$-cobordisms between them. Because the theory has a combinatorial definition, all cohomology groups and maps between them will be algorithmically computable. If successful, this program will realize the Jones polynomial as the Euler characteristic of a cohomology theory of link cobordisms.

Section 8 presents mild variations on cohomology groups $H^{i}(D)$ and ${\cal H}^{i,j}(D).$ There we switch from links to $(1,1)$-tangles. We consider the category $A\mbox{-mod}_{0}$ of graded $A$-modules and grading-preserving homomorphisms between them. Given a plane diagram $D$ of a $(1,1)$-tangle $L$ and a graded $A$-module $M,$ in Section 8.3 we define cohomology groups $H^{i}(D,M)$ which are graded $A$-modules. The arguments of Sections 4-5 go through without a single alteration and show that isomorphism classes of $H^{i}(D,M)$ do not depend on the choice of $D$ and are invariants of the underlying $(1,1)$-tangle $L$. In fact, to every $(1,1)$-tangle and an integer $i$ we associate an isomorphism class of functors from the category of graded $A$-modules to itself.

Motivations for this work and its relations to representation theory. What is the representation-theoretical meaning of the cohomology groups $H^{i,j}(D)?$ The Jones polynomial of links is encoded in the finite-dimensional representation theory of the quantum group $U_{q}(\mathfrak{sl}_{2}).$ It was shown in [FK] and [K] that the integrality and positivity properties of the Penrose-Kauffman $q$-spin networks calculus, of which the Jones polynomial is a special instance, are related to Lusztig canonical bases in tensor products of finite-dimensional $U_{q}(\mathfrak{sl}_{2})$-representations. Lusztig’s theory [L], among other things, says that various structure coefficients of quantum groups can be obtained as dimensions of cohomology groups of sheaves on quiver varieties. This suggests a “categorification” of quantum groups and their representations, i.e. that there exist certain categories and 2-categories whose Grothendieck groups produce quantum groups and their representations.

Louis Crane and Igor Frenkel [CF] conjectured that quantum $\mathfrak{sl}_{2}$ invariants of 3-manifolds can be lifted to a 4-dimensional topological quantum field theory via canonical bases of Lusztig. They also introduced a notion of Hopf category and associated to it 4-dimensional invariants. Representations of a Hopf category form a 2-category, and a relation between 2-categories and invariants of 2-knots in $\mathbb{R}^{4}$ was established in [Fs].

In joint work with Bernstein and Frenkel [BFK], we propose a categorification of the representation theory of $U_{q}(\mathfrak{sl}_{2})$ via categories of highest weight representations for Lie algebras $\mathfrak{gl}_{n}$ for all natural $n.$ This approach can be viewed as an algebraic counterpart of Lusztig’s original geometric approach to canonical bases. Motivated by the geometric constructions of [BLM] and [GrL], we obtain a categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}_{2})$ via projective and Zuckerman functors. We consider categories $\mathcal{O}_{n}$ that are direct sums of certain singular blocks of the category $\mathcal{O}$ for $\mathfrak{gl}_{n}.$ Given a tangle $L$ in the 3-space with $n$ bottom and $m$ top ends and a plane projection $P$ of $L,$ we associate to $P$ a functor between derived categories $D^{b}(\mathcal{O}_{n})$ and $D^{b}(\mathcal{O}_{m}).$ Properties of these functors suggest that their isomorphism classes, up to shifts in the derived category, are invariants of tangles. When the tangle is a link $L,$ we expect to get cohomology groups $\mathbb{H}^{i}(L)$ as invariants of links. These groups will be a special case of the cohomology groups constructed in this paper: conjecturally

Acknowledgements. This work was started during a visit to the Institut des Hautes Etudes Scientifiques and finished at the Institute for Advanced Study. I am grateful to these institutions for creating a wonderful working atmosphere. During my stay at the Institute for Advanced Study I was supported by grant DMS 9304580 from the NSF.

I am indebted to Joseph Bernstein for interesting discussions, to Greg Kuperberg for reading and correcting the first version of the manuscript and explaining to me a natural way to hide minus signs, and to Oliver Dasbach and Arkady Vaintrob for pointing out that Corollary 13 was proved by Thistlethwaite [T].

On numerous occasions Igor Frenkel, who was my supervisor at Yale University, advised me to look for a lift of the Penrose-Kauffman quantum spin networks calculus to a calculus of surfaces in $\mathbb{S}^{4}.$ This work can be seen as a partial answer to his questions. It is a pleasure to dedicate this paper to my teacher Igor Frenkel.