is twisted torus knot T(7,2;4,1) prime? what kind of twisted torus knot is prime? 
even more , is twisted torus knot T(7,2;4,1) prime?
 A: Yes, a positive twisted torus knot is always prime, proved by the fact that it is a Lorenz knot [Corollary 1, "A new twist on Lorenz links" by Birman-Kofman].  In that paper, this knot is called T((4,4),(7,2)), but in the paper referenced by Sam Nead, it is called T(7,2,4,4). 
Here is some more information about this knot:  It is fibered with genus 9.  Its crossing number is 21, and its braid index is 4.  It is not hyperbolic and not T(4,7), so it must be a satellite knot.
To justify these facts, we compute its Jones polynomial: 
t^(9) + t^(11) + t^(13) - t^(14) + t^(15) - 2t^(16) + t^(17) - 2t^(18) + 2t^(19) - t^(20) + t^(21) - t^(22).
Thus, genus g=9.  Its braid index n=min(s+q,r)=4.  Then using 2g=c-n+1, we get c=21.
Snappy tells us that this knot is not hyperbolic, and we can rule out T(4,7) using the Jones polynomial.
A: Twisted torus knots are mostly hyperbolic, and in particular prime.  The following paper can serve as an introduction to the literature.
http://arxiv.org/abs/1007.2932
