I have a specific knot $K$ (it seems that it is $6_3$) and want to find its braid index. I managed to construct a braid with 3 strings whose closure is $K$, however I do not know whether 2 strings would be enough.
The only observation I did is that if there exists such a braid with 2 strings, then it must have odd number of crossings, since otherwise we will get a link that is not a knot.
Any ideas on how to check whether a braid with 2 strings can work?
Thank you.
On
The knots that have 2-string braids are a very concrete and well understood class, they are precisely the $(2,n)$ torus knots with $n$ odd, e.g. the trefoil knot also known as the $(2,3)$ torus knot, and see this list which depicts the next few $(2,5)$, $(2,7)$, and $(2,9)$, and a somewhat Celtically stylized depiction of $(2,11)$.
So much concrete information is known about torus knots in general and the $(2,n)$ torus knots in particular that you should be able to easily distinguish them from other knots. For example, the complement of any torus knot is a Seifert fibered 3-manifold hence the center of the corresponding knot group is an infinite cyclic group (in fact I believe that the torus knots are the only knots whose group has nontrivial center).
To expand on Lee Mosher's answer, $(2,q)$-torus knots must have Alexander polynomial of the form $\frac{t^q+1}{t+1}$, which means the coefficients have to alternate between $\pm 1$. On the other hand, according to the Knot Atlas (http://katlas.org/wiki/6_3), knot $6_3$ has Alexander polynomial $t^2-3t+5-3t^{-1}+t^{-2}$, so it cannot be a $(2,q)$-torus knot.