give me an example in detail on how to obtain the Khovanov homology from the Jones polynomial

The Khovanov homology is a powerful tool in knot theory that provides a more refined invariant of knots than the Jones polynomial. In this example, we will show how to obtain the Khovanov homology of a knot from its Jones polynomial.

Let us consider the trefoil knot, which has the Jones polynomial:

$$V(q) = q^{-1} + 1 - q$$

To obtain the Khovanov homology of the trefoil knot, we need to construct a two-dimensional grid, where the rows represent the grading of the homology and the columns represent the grading of the cohomology. We will label the rows and columns with integers, starting from 0.

For the trefoil knot, the grid looks like this:

    -2 -1  0  1  2     
   +----------------
 0 |   
 1 | 
 2 | 
 3 |
 4 |
 5 |
 6 |
   +----------------

We will now fill in the entries of the grid using the following algorithm:

1. Start with the row labeled 0. Set the entry in the column labeled 0 to be 1.

2. For each row labeled $i$, where $i$ is odd, compute the entry in the column labeled $j$ using the formula:

$$b_{i,j} = q^{j-i}b_{i-1,j} + q^{j-i+2}b_{i-1,j-1}$$

where $b_{i,j}$ denotes the entry in row $i$ and column $j$.

3. For each row labeled $i$, where $i$ is even, compute the entry in the column labeled $j$ using the formula:

$$b_{i,j} = q^{j-i+1}b_{i-1,j} + q^{j-i+1}b_{i-1,j-1}$$

4. Repeat steps 2 and 3 until all rows have been filled.

Applying this algorithm to the trefoil knot, we obtain the following grid:

    -2 -1  0  1  2     
   +----------------
 0 |  1   
 1 |  0  q  0 
 2 |  q  0  2q^{-1}  0  q 
 3 |  0  q  0  -q^{-1}  0 
 4 |  q^{-1}  0  q^{-1}  0  q^{-1} 
 5 |  0  -q^{-1}  0  q  0 
 6 |  -q  0  -2q  0  -q 
   +----------------

The Khovanov homology of the trefoil knot is obtained by taking the graded Euler characteristic of this grid. The graded Euler characteristic is defined as:

$$\sum_{i,j} (-1)^i b_{i,j}$$

which in this case evaluates to:

$$1 - q^2 + q^4$$

Thus, the Khovanov homology of the trefoil knot is the vector space with basis elements labeled by the integers $-2,-1,0,1,2$ and with the following gradings:

$$\begin{aligned} \mathrm{Kh}^{-2} &= \mathbb{F} \ \mathrm{Kh}^{-1} &= \mathbb{F}^2 \ \mathrm{Kh}^{0} &= \mathbb{F}^3 \ \mathrm{Kh}^{1} &= \mathbb{F}^2 \ \mathrm{Kh}^{2} &= \mathbb{F} \end{aligned}$$

where $\mathbb{F}$ is the field with two elements. The differential in the Khovanov complex is given by the matrix:

$$\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 \ 0 & q & 0 & 0 & 0 \ 0 & 0 & -1 & 0 & 0 \ 0 & 0 & 0 & q^{-1} & 0 \ 0 & 0 & 0 & 0 & -1 \end{matrix} \right)$$

which induces the Khovanov homology groups:

$$\begin{aligned} \mathrm{Kh}^{-2} &= \mathbb{F} \ \mathrm{Kh}^{-1} &= \mathbb{F}^2 \ \mathrm{Kh}^{0} &= \mathbb{F}^3 \ \mathrm{Kh}^{1} &= \mathbb{F}^2 \ \mathrm{Kh}^{2} &= \mathbb{F} \end{aligned}$$

with trivial differentials.