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We introduce a simple recursive relation and give an explicit formula of the Kauffman bracket of two-strand braid link . Then, we give general formulas of the bracket of the sequence of links of three-strand braids . Finally, we give an interesting result that the Kauffman bracket of the three-strand braid link is actually the product of the brackets of the two-strand braid links and . Moreover, a recursive relation for is also given.

The Kauffman bracket polynomial was introduced by L. H. Kauffman in 1987 [

The Kauffman bracket (polynomial) is actually not a link invariant because it is not invariant under the first Reidemeister move. However, it has many applications and it can be extended to a popular link invariant, the Jones polynomial. In the present work we shall confine ourselves to the Kauffman bracket to avoid this work from unnecessary length and to leave it for applications.

This paper is organized as follows: In Section 2 we shall give the basic ideas about knots, braids, and the Kauffman bracket. In Section 3 we shall present the main results.

A link is a disjoint union of circles embedded in

Two links are isotopic if and only if one of them can be transformed to the other by a diffeomorphism of the ambient space onto itself. A fundamental result by Reidemeister [

The set of all links that are equivalent to a link

The main question of knot theory is Which two links are equivalent and which are not? To address this question one needs a knot invariant, a function that gives one value on all links that belong to a single class and gives different values (but not always) on knots that belong to different classes. The present work is basically concerned with this question.

Braids were first studied by Emil Artin in 1925 [

An n-strand

The product

A braid with only one crossing is called elementary braid. The ith elementary braid

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

An important result by Alexander [

Remark 2.1 In the last section, all the concerned links will be closures of products of elementary braids.

Before the definition it is better to understand the two types of splitting of a crossing, the A-type and the B-type splittings:

In the following, the symbols

Definition 2.2 The Kauffman bracket is the function

Here

Proposition 2.3 The Kauffman polynomial is invariant under second and third Reidemeister moves but not under the first Reidemeister move [

In this section we shall introduce a recursive relation for the Kauffman bracket, shall give an explicit formula of

First of all we give the Kauffman bracket of the

Lemma 3.1 The Kauffman bracket of the

Proof. We prove it by induction on

The case

Theorem 3.2 (A recursive relation) The following relation holds for any

Proof. We prove it using directly the definition and Lemma 3.1:

From this recursive relation, we get the explicit formula for the 2-strand braid link

Proposition 3.3 The Kauffman bracket of the link

Proof. We prove it by induction on

For

which satisfies the recursive relation.

With the assumption that the relation holds for an arbitrary n, we, using Theorem 3.2, get

This completes the proof. ,

In the following we give the Kauffman bracket polynomial of the closure of the braid

Proposition 3.4 The Kauffman bracket of

Proof. Simply, apply the definition for different values of

Lemma 3.5 The Kauffman brackets for

Proof. The proofs of first three cases are given (proofs of remaining cases are similar):

Theorem 3.6 For any

Proof. We prove it by induction on

For instance,

In connected sum

Lemma 3.7

Proof. We prove it by induction on

For

Now, with the assumption that the result holds for an arbitrary

as required. ,

The following result confirms that the Kauffman bracket of

Theorem 3.8 For any

Proof. We prove it by induction on

When

Suppose the result holds for

Now, using Lemma 3.7, we have

This completes the proof. ,

Corollary 3.9

Proof. It is obvious:

Corollary 3.10

and

Proof. The result follows immediately from Theorem 3.8 as

For the following, let us fix the notation

and

Proposition 3.11 The Kauffman bracket of the link

Proof. We prove it by induction on

For

Now, with the assumption that the result holds for an arbitrary

as required. ,

Proposition 3.12 The Kauffman bracket of the link

Proof. We prove it by induction on

For

Now, with the assumption that the result holds for

as was required. ,