Are all torus knots rational knots (2 bridge knots?)
Also, does anyone know where I can find an explcit formula for the Jones / Alexnader / HOMFLY polynomial for Torus knots??? What about when they are of type (2,2k+1)?
I'd really appreciate it!! Thanks!!
No, there are torus knots which are not rational knots. The only torus knots which are 2-bridge knots are of the form $T(2,n)$, which is of course, the same as $T(n,2)$. These are the same as the rational knot associated to $\frac{1}{n}$.
Jones himself was the first to compute the Jones polynomial for torus knots. Link here. Look at the very end of that pdf to see that for a $T(p,q)$ torus knot, the Jones polynomial is
$$ \frac{t^{ \frac{(p-1)(q-1)}{2} }}{1-t^2} (1−t^{p+1}−t^{q+1}+t^{p+q}) $$
For the others, I am not sure if there are general formula, but Wolfram has a recurrence relation for the Bracket polynomial of torus knots $T(2,n)$, so you should be able to apply to your case in some way, but this is closer to the Jones polynomial than the other two.
But $T(2,2k+1)$ is a pretty nice class of knots, so I wouldn't be surprised if you dig around you can find more about this class of knots' polynomials.