# Khovanov, Categorification 

:::{.remark}
CS functional critical for 3 and 4 dimensional TQFTs.
In 3d, path integral in Witten's construction of WRT invariants.
In 4d, in instanton Floer homology.
Generally a tensor functor from 3-manifolds with 4d cobordisms to some algebraic category like $\Ab\Grp$.
Motivational problem: construct a 4D TQFT that categorifies the WRT invariant for 3-manifolds.
Look at tensor functors from the categories of links in $\RR^3$ and link cobordisms in $\RR^3\cross I$ to some algebraic category.

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/999_Chern/sections/figures/2021-11-18_14-35.xoj -->

\begin{tikzpicture}
\fontsize{44pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/999_Chern/sections/figures}{2021-11-18_15-43.pdf_tex} };
\end{tikzpicture}

:::

:::{.remark}
Components in links colored by irreps of a simple Lie algebra $\lieg$.
Categorifying: recover polynomials as graded Euler characteristics.
In some cases, not clear how to extend link homology to link cobordisms.
Link homology is best understood with miniscule representations, e.g. $\lieg = \liesl_n$, labeled by $\Extalg^k V$ for $V\cong \CC^N$ the fundamental representation.
For various $N$, recovers Jones, Alexander, HOMFLYPT polynomials.
Skein relation: assign a complex to resolutions, find an exact triangle, set the original complex to be the cone of the map between the complexes of the two resolutions.
Why this is a good idea: reduces 3d diagram projections to actual 2d planar graphs.
:::

:::{.remark}
WRT is an invariant of a planar graph, homology supported in a single degree.
Maps between graph homology induced by foams.
Look for functors that assign objects to vector spaces and foams to morphisms.
A foam: for $\SL_3$, a 2d combinatorial CW complex with generic singularities embedded in $\RR^3$.
Examples:

- $\Sigma_{g, n}$
- $S^2$ with a marked point and an equatorial disc attached.
- $S^1\times S^1$ with an equatorial disc.

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/999_Chern/sections/figures/2021-11-18_15-36.xoj -->

![](figures/2021-11-18_15-42-39.png)

Introduce Tait coloring: any two facets sharing an edge must have distinct colors.
E.g. the sphere above admits 6 such colorings over $\FF_3$, since the 3 facets (north/south hemispheres and equatorial disc) share an edge (equator).
Theorem: can produce a closed orientable surface in $\RR^3$ from this.
The procedure is roughly numbering the facets, then taking $F_{ij}$ to be the surface obtained by leaving $\ts{i,c}^c$ facets out.
E.g. for index set $\ts{1,2,3}$, you get $F_{12}, F_{13}, F_{23}$.
Construct an evaluation map by constructing local evaluations with respect to an admissible coloring, then sum over such colorings to get a symmetric function in $\kxn^{S_n}$.
Initially defined to be rational, i.e. $\gens{F} \in k(x_1,x_2,x_3)$, but it turns out that denominators cancel and $\gens{F} \in k[x_1,x_2,x_3]^{S_3}$.
Can pair foams by flipping and gluing along boundary to get a closed foam, then apply evaluation.
Yields a bilinear pairing, consider the kernel to get interesting skein-type relations.
:::

:::{.remark}
General construction yields a lax tensor functor: just a map, not an isomorphism.
:::

:::{.remark}
Conjecture: the state space $\gens{\Gamma}$ for $\Gamma$ a graph is a free \(R\dash\)module of rank $r$ the number of Tait colorings.
Known for reducible graphs, i.e. skein relations can eventually break them into empty graphs.
Theorem is true after a certain modification to $\gens{\Gamma}_\phi$?
Motivations coming from Kronheimer and Mrowka 2019, studying $\SO_3$ instanton Floer homology over $\FF_2$ for 3-orbifolds using a CS functional for orbifolds, primarily $\RR^3/(C_2\cartpower{2})$ to get a trivalent vertex in a graph.
Yields a homology theory for trivalent graphs in $\RR^3$.
Might give a new way to think about the four color theorem!
Reductions: reduce links in $\RR^3$ to diagrams in $\RR^2$, set up evaluation, then take cones?
:::

:::{.remark}
Current work: how to categorify $\zeta^p = 1$?
Possibly use cyclotomic rings in characteristic $p$.
:::