^{1}

^{1}

^{2}

^{1}

Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all of them. We give the graded Euler characteristic and the Khovanov homology of the 2-strand braid link ,, and the 3-strand braid .

Khovanov homology is an invariant for oriented links which was introduced by Mikhail Khovanov in 2000 as a categorification of the Jones polynomial [

Khovanov assigned a bigraded chain complex

Although Khovanov’s construction is combinatorial from which Khovanov homology is algorithmically computable, we shall follow rather a simple way of Bar-Natan’s, which he introduced in [

A link in

Links are usually studied via projecting them on the plane. A projection with information of over- and under- crossing is called a link diagram. Some link diagrams are given in

Two links are called isotopic (or equivalent) if one of them can be transformed to another by a diffeomorphism of the ambient space

Remark 1. By a link we shall mean a diagram of its isotopy class.

Reidemeister gave in [

To classify links one needs a link invariant [

An n-strand braid is a set of n non-intersecting smooth paths connecting n points on a horizontal plane to n points exactly below them on another horizontal plane in an arbitrary order [

The product ab of two n-strand braids is defined by putting the braid a above the braid b and then gluing their common end points. A braid with only one crossing is called elementary braid. The ith elementary braid x_{i} on n strands is given in

A useful property of elementary braids is that every braid can be written as a product of elementary braids. For instance, the above 2-strand braid is

The closure of a braid b is the link

Theorem 1. [

In 1985 V. F. R. Jones revolutionized knot theory by defining the Jones polynomial as a knot invariant via Von Neumann algebras [

A Kauffman state s of a link L is obtained by replacing each crossing (

Let s be a state in

It is well known that the Kauffman bracket satisfies the relations:

This bracket is not invariant under the first Reidemeister move [

and its normalized version by the relation

Since this polynomial is invariant under all three Reidemeister moves, it is an invariant for oriented links.

Example 1. It is easy to check that the normalized Jones polynomial of the link

Definition 1. A graded vector space W is a decomposition of W into a direct sum of the form

where each

Definition 2. Let V and W be two homogeneous components of graded vector spaces. The degree of the tensor product

Definition 3. Let

Definition 4. The degree shift

Definition 5. Bar-Natan discovered in [

and

toremovenumbering (beforeeachequation)

Here V is a graded vector space with graded dimension

Definition 6. The chain complexample

The height shift operation

Definition 7. The graded Euler characteristics of a chain complexample is defined to be the alternating sum of the graded dimensions of its homology groups, i.e.

Theorem 2. [

Theorem 3. [

Now we give the graded Euler characteristic of

Before computing the Khovanov homology, we define two gradings, the homological grading and the quantum grading. The homological grading of the chain complexample is defined as

Example 2. Here is the Khovanov homology of

1) The n-cube: The 3-cube of the trefoil knot

2) Khovanov Bracket: The Khovanov brackets along with their q-dimensions are given in

3) Unnormalized Jones polynomial: The graded Euler characteristic of

4) Khovanov Homology: In order to compute the Khovanov homology of

The Khovanov Homology of the link

Khovanov Bracket | q-dimension |
---|---|

Homology degree | |||||
---|---|---|---|---|---|

3 | 2 | 1 | 0 | ||

Grading | 1 | ||||

3 | |||||

5 | |||||

7 | |||||

9 |

Remark 2.

This section contains the chain complex, Khovanov bracket, graded Euler characteristic, and Khovanov homology of the braid link

Proposition 4. The chain complex of the link

Proof. We proof it by induction on n, using the trick that instead of “®”, we use “+” and that instead of

The expansion holds obviously for n = 1, that is

Now, suppose that the result holds for n = k, that is

For n = k + 1, we have

Now, replacing 1 by

Theorem 5. The graded Euler characteristic of

Proof. The proof is simple; just by following the definition. □

Proposition 6. The unnormalized Jones polynomial of

and the normalized is

Proof. Since the unnormalized Jones polynomial is the alternative sum of Khovanov brackets, we have

Now after cancelation of terms, which behave differently for even and odd n, we receive the desired result.

For instance, see the cases for n = 5, 6:

Theorem 7. (Main theorem) a) If n is even, then

b) If n is odd, then

c) If

Proof. We prove it using the relation

and establishing a table with the help of the quantum and homological gradings. The homological grading r appears in a row and quantum grading q appears in a column. The homological gradings receive alternating signs, starting positive sign from 0; a term with negative sign appears at an odd r, while the positive sign appears at an even r. The powers of q in the relation represent the quantum grading. Corresponding to each term in the relation, a

a) In case of even number of crossings we receive a 2-component link; hence, at n^{th} homological grading, two

b) However, in odd number of crossing we always receive a knot; this confirms that at highest homological grading there exists a ^{th}. Moreover, at

c) Since at height 0 we receive the space^{th} homological level there exist two ^{th} quantum gradings. This completes the proof. □

Now we give the graded Euler characteristic of the 3-strand braid

Theorem 8.

1)

2)

3)

4)

∙∙∙ | − | + | − | + | ||||
---|---|---|---|---|---|---|---|---|

n | n − 1 | n − 2 | ∙∙∙ | 3 | 2 | 1 | 0 | |

n − 2 | ||||||||

n | ||||||||

n + 2 | ||||||||

3n |

∙∙∙ | − | + | − | + | ||||
---|---|---|---|---|---|---|---|---|

n | n − 1 | n − 2 | ∙∙∙ | 3 | 2 | 1 | 0 | |

n − 2 | ||||||||

n | ||||||||

n + 2 | ||||||||

3n |

X | Y | |
---|---|---|

5)

6)

Proof. (4) Since there are 6k + 3 crossings in the link

The result now follows using the definition and simplifying the expression.

See, for example, the case for k = 1. The figure on the right represent the link of the reduced form of Δ^{3},

which is

Level | Khovanov Bracket | q-dimension |
---|---|---|

0 | ||

1 | ||

2 | ||

3 | ||

Level | Khovanov bracket | q-dimension |
---|---|---|

0 | ||

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 |

Homological grading q | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | ||

Quantum grading r | 6 | ||||||||||

8 | |||||||||||

10 | |||||||||||

12 |

For Khovanov brackets and q-dimensions for smoothings of

The proofs of other parts are similar to the proof of Part 4. □

Abdul Rauf Nizami,Mobeen Munir,Tanweer Sohail,Ammara Usman, (2016) On the Khovanov Homology of 2- and 3-Strand Braid Links. Advances in Pure Mathematics,06,481-491. doi: 10.4236/apm.2016.66034