**Remark 3.**


An important result by Alexander, connecting links and braids, is: **Theorem 2.** (**Alexander** [21]). *Each link is a closure of some braid.*

**Definition 9.** *The* 0*- and* 1*-smoothings of crossing are defined, respectively, by and* .

**Definition 10.** *A collection of disjoint circles obtained by smoothing out all the crossings of a link L is called the Kauffman* state *of the link [22].*
