*Note:
*

*These are notes live-tex’d from a graduate course in Categorification taught by Ariki Wilbert at the University of Georgia in Summer 2020. As such, any errors or inaccuracies are almost certainly my own.*

dzackgarza@gmail.com

Last updated: 2020-10-25

We’ll start with \(X\) a finite CW complex.

A **CW complex** is a topological space built by inductively attaching \(i{\hbox{-}}\)dimensional discs (\(i{\hbox{-}}\)cells) \({\mathbb{D}}^i \mathrel{\vcenter{:}}=\left\{{\mathbf{x}\in {\mathbb{R}}^i {~\mathrel{\Big|}~}{\left\lVert {\mathbf{x}} \right\rVert} \leq 1}\right\}\) along their boundary \({{\partial}}{\mathbb{D}}^i = S^{i-1} \mathrel{\vcenter{:}}=\left\{{\mathbf{x}\in {\mathbb{R}}^i {~\mathrel{\Big|}~}{\left\lVert {\mathbf{x}} \right\rVert} = 1}\right\}\).

Define \(\chi(X) = \sum_{i\in {\mathbb{Z}}} (-1)^i {\left\lvert {C_i} \right\rvert}\) where \({\left\lvert {C_i} \right\rvert}\) is the number of \(i{\hbox{-}}\)cells.

Note that a homotopy equivalence between spaces induces an equality between Euler characteristics.

Recall that we can define the cellular chain complex \[\begin{align*} C_*^{\text{cell}}(X, {\mathbb{C}})\mathrel{\vcenter{:}}=\cdots \xrightarrow{{\partial}_{i+1}} C_n^{\text{cell}} (X, {\mathbb{C}}) \xrightarrow{{\partial}_{i}} \cdots \to C_0^{\text{cell}}(X, {\mathbb{C}}) \end{align*}\]

and \(H_i(X, {\mathbb{C}}) \mathrel{\vcenter{:}}=\ker {\partial}_i / \operatorname{im}({\partial})_{i+1}\).

\[\begin{align*} \sum (-1)^i \dim H_i(X, {\mathbb{C}}) = \chi(X) \end{align*}\]

In this sense, cellular homology categorifies the Euler characteristic: we’ve replaced a set of objects with a category. This is an improvement because we may not have maps between the elements of sets, *but* we do have maps between objects in a category. We can also talk about things such as functoriality.

The euler characteristic is a weaker invariant than homology. Note that \[\begin{align*} \chi(S^1) = 0 &\quad\text{and}\quad \chi(S^1{\coprod}S^1) = 0 \\ \\ H_0(S^1) = {\mathbb{C}}&\quad\text{while}\quad H_0(S^1{\coprod}S^1) = {\mathbb{C}}\oplus {\mathbb{C}} ,\end{align*}\] so these aren’t distinguished by euler characteristic alone.

Our first goal will be to assign invariants to oriented links \(L\), where homotopy equivalence will be replaced with isotopy. We’ll assign a Khovanov complex \(C_*(L,{\mathbb{C}})\), a complex of \({\mathbb{Z}}{\hbox{-}}\)graded \({\mathbb{C}}{\hbox{-}}\)vector spaces, along with the Jones polynomial \(J(L) \in {\mathbb{Z}}[t, t^{-1}]\). By taking the (graded) Euler characteristic of the chain complex, we’ll recover \(J(L)\).

A *link* \(L\) is a smooth, closed 1-dimensional embedded submanifold of \({\mathbb{R}}^3\). \(L\) is a *knot* if it consists of one connected component.

We have planar projections:

Under this correspondence, isotopy of knots will correspond to planar isotopy of the diagrams and the following 3 Reidemeister moves:

There are three planar moves that preserve the isotopy class of a planar projection of a knot:

How to change knot diagrams using Reidemeister moves:

We now want to take an oriented, planar link diagram \(D\) and associate to it a polynomial \(J(D)\). We start by defining the Kauffman bracket

Let \(D_f\) be \(D\) with the orientation forgotten, then \(\left\langle{D_f}\right\rangle \in {\mathbb{Z}}[v, v^{-1}]\) is defined recursively by

In the last case, the first term is a “0-resolution/0-smoothing” and the second is a “1-resolution/1-smoothing”.

We have a notion of positive/negative crossings:

We set \[\begin{align*} J(D) = (-1)^{n_-} v^{n^+ - 2n_-} \left\langle{D_f}\right\rangle .\end{align*}\]

\(J(S^1) = v + v^{-1}\)

\(J(?) = (-1) v^{-2} \qty{ -v^2 (v+v^{-1})} = v+v^{-1}\).

\(J(?) = v^{-6} + v^{-4} + v^{-2} + 1\)

The Jones polynomial is invariant under move \(R1\).

Can be checked in diagrams:

You can now check that \[\begin{align*} J(D') = (-1)^{n_-(D)} v^{n_+(D) + 1} - 2n_- .\end{align*}\]

Check invariance under R2, R3.

Recall that we had recursive rules for computing the Kausffman bracket, and a normalization factor for the Jones polynomial that made it into an invariant. We’d like a closed formula for these.

We do this by ordering the crossings of the unoriented link \(1, \cdots, n\), then there is a correspondence \[\begin{align*} \left\{{0, 1}\right\}^n &\iff \text{Complete resolutions} \\ (\alpha_1, \cdots, \alpha_n) &\iff \alpha_i \text{ resolves the $i$th crossing} .\end{align*}\]

\[\begin{align*} \left\langle{D}\right\rangle = \sum_{\alpha \in \left\{{0, 1}\right\}^n} (-1)^{{\left\lvert {\alpha} \right\rvert}} v^{{\left\lvert {\alpha} \right\rvert}} (v+v^{-1})^{c_\alpha(D)} ,\end{align*}\]

where \({\left\lvert {\alpha} \right\rvert}\) is the number of 1-resolutions and \(c_\alpha\) is the number of circles in the resolution corresponding to \(\alpha\).

Idea: look at resolving the \(n\)th crossing locally and apply the recursive relation. Then rewrite the sum by appending \(\alpha_n = 0\) and \(\alpha_n = 1\) respectively. Note that we can rewrite the sum as \[\begin{align*} \sum_{r=0}^n (-1)^r \sum_{{\left\lvert {\alpha } \right\rvert}= r} v^r (v+v^{-1})^{c_\alpha(D)} .\end{align*}\]

This amounts to summing over the “columns” in the previous diagram:

Here this yields \[\begin{align*} (v+v^{-1})^2 + (-1)2v(v+v^{-1}) + v^2 (v+v^{-1})^2 .\end{align*}\]

Note that this formula starts to resemble an Euler characteristic!

*Problem*: The coefficient \[\begin{align*} \sum v^r(v+v^{-1})^{c_\alpha(D)} \in {\mathbb{Z}}^{\geq 0}[v, v^{-1}] \end{align*}\] is a Laurent polynomial instead of a natural number, so this can’t immediately be interpreted as a dimension of a vector space.

*Solution*: Replace finite-dimensional \({\mathbb{C}}{\hbox{-}}\)vector spaces by \({\mathbb{Z}}{\hbox{-}}\)graded vector spaces. The category consists of objects given by \(V = \bigoplus_{i\in {\mathbb{Z}}} V_i\) and linear maps \(f:V\to W\) such that \(f(V_i) \subseteq W_i\) for all \(i\).

We previously had vector spaces categorifying the natural numbers by taking the dimension, so for graded vector spaces, we take the **graded dimension**:

\[\begin{align*} {\operatorname{gr\,dim}}\bigoplus_{i\in{\mathbb{Z}}}V_i = \sum_{i\in {\mathbb{Z}}}\qty{\dim V_i}v^i \in {\mathbb{Z}}^{\geq 0}[v, v^{-1}] .\end{align*}\]

**Goal**: We want to associate to an oriented link diagram \(D\) a cochain complex of finite-dimensional graded \({\mathbb{C}}{\hbox{-}}\)vector spaces \(C_i(D) \xrightarrow{{{\partial}}} C_{i+1}(D) \to \cdots\). Since each chain space decomposes, the differential does as well, and we get a large collection of chain complexes