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We give a general formula of the quantum sl_{2}-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra g=sl_{2} in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum invariant.

The discovery of the Jones polynomial inspired many people to search for other skein relations compatible with Reidemeister moves and thus defined knot polynomials. This led to the introduction of the HOMFLY and Kauffmans polynomials. It soon became clear that all these polynomials are the first members of a vast family of knot invariants called quantum invariants.

The original idea of quantum invariants was proposed by E. Witten in [

This paper is organized as follows: In Section 2, we give the basic ideas about knots, tangles, the Jones polynomial, Lie algebra representations, and construction of quantum invariants. In Section 3, we present the main result along with its specialization to the Jones polynomial.

A knot is a circle embedded in

A crossing trivial knot trefoil knot Two knots are called isotopic if one of them can be transformed to the other by a diffeomorphism of the ambient space

Two unoriented knots

R1 R2 R3 The set of all knots that are equivalent to a knot

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 knots that belong to a single class and gives different values (but not always) on knots that belong to different classes. The present work is concerned with this question.

A tangle is a generalization of a knot which at the same time is simpler and more complicated than a knot. On one hand, knots are a particular case of tangles, on the other hand, knots can be represented as combinations of (simple) tangles.

A tangle in a knot projection is a region in the projection plane surrounded by a circle such that the knot crosses the circle exactly at four places.

A tangle The following two operations are defined on tangles: When the bottom of a tangle

The second operation, tensor product, is defined by placing one tangle next to the other tangle (of the same height).

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

Definition 1 [7-9] The Jones polynomial

and that the value of the unknot is 1. Here

For instance, it is easy to verify that the Jones polynomial of the left-handed trefoil knot (which is denoted by

Let

An important property of quantum groups is that every representation gives rise to a solution

where

In case of Lie algebra

for an appropriate basis

The general procedure of constructing quantum invariants is as follows (see details in [

A portion of a knot diagram between two such horizontal lines represents a tangle

·

·

·

Now we can define a knot invariant

Because of the multiplicity property

Using this one can verify that

To complete the construction of our quantum invariant we should assign appropriate operators to the minimum and maximum points. These depend on all the data involved: the quantum group, the representation and the

where

In the following example we compute the quantum

Example 1 Let us compute the

So,

Here we give the general formulas of the quantum

Proposition 1 The quantum invariant of

Proof 1 We prove it by induction on

For

Note that the map

Now applying

Finally, the two maps at the top contract the whole tensor into the linear transformation

Hence the unframed normalized

To get a clear picture, we also compute the quantum invariant of the knots

The map at the bottom sends 1

Now the map

Then applying

Finally, the two maps at the top contract the whole tensor into a number

Dividing by the normalizing factor

The invariant

where

This can be further written as

For

With some computations, similar to the computations of

Similarly,

We now assume the result (1.1) holds for

Now for

and the proof is finished.

Proposition 2 The Jones polynomial of the knot

Proof 2 Nothing to prove; just substitute