**2. Braid Links**

**Definition 1.** *A* knot *is a simple, closed curve in the three-space. More precisely, it is the image of an injective, smooth function from the unit circle to* R<sup>3</sup> *with a nonvanishing derivative [18]. You can see some knots in Figure 1*:

**Definition 2.** *An m-component* link *is a collection of m nonintersecting knots [18]. A trivial two-component link and the Hopf link are given in Figure 2:*

**Figure 2.** Links.

**Definition 3.** *Two links L*1 *and L*2 *are said to be* isotopic *or* equivalent *if there is a smooth map F:* [0, 1] × *S*1 → <sup>R</sup>3*, which confirms that Ft is a link for all t* ∈ [0, 1] *and that that F*0 = *L*1 *and F*1 = *L*2*. Map F is called* isotopy*. By the isotopy class of a link L, denoted* [*L*]*, we mean the collection of all links that are isotopic to L.*

Since it is hard to work with links in R3, people usually prefer working with their projections on a plane. These projections should be generic, which means that all multiple points are double points with a clear information of over- and undercrossing, as you can see in Figure 3. Such a projection of a link is called the *diagram* of the link.

**Theorem 1.** (**Reidemeister**, [19]). *Let D*1 *and D*2 *be two diagrams of links L*1 *and L*2*. Then, links L*1 *and L*2 *are isotopic if and only if D*1 *is transformed into D*2 *by planar isotopies and by a finite sequence of three localmovesrepresentedinFigure4:*

**Figure 4.** Reidemeister moves.

**Definition 4.** *A* link invariant *is a function that remains constant on all elements in an isotopy class of a link.*

**Remark 1.** *A function to qualify as a link invariant should be invariant under the Reidemeister moves.*

**Definition 5.** *An n-strand braid is a collection of n nonintersecting, smooth curves joining n points on a plane to n points on another parallel plane in an arbitrary order such that any plane parallel to the given planes intersects exactly n number of curves [20]. The smooth curves are called the strands of the braid. You can see a 2-strand braid in Figure 5:*

**Definition 6.** *The* product *of two n-strand braids α and β, denoted by αβ, is defined by putting β below α and then gluing their common endpoints.*

**Remark 2.** *Each braid is a product of elementary braids.*

**Definition 8.** *The* closure *of a braid β, denoted by β* - *, is defined by connecting its lower endpoints to its corresponding upper endpoints with smooth curves, as you can see in Figure 7.*

**Figure 7.** Braid closure.

