what kind of twisted torus knot is prime? even more , is twisted torus knot T(7,2;4,1) prime?

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 BirmanKofman]. 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=cn+1, we get c=21. 


Twisted torus knots are mostly hyperbolic, and in particular prime. The following paper can serve as an introduction to the literature. 

