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

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What definition of twisted are you using? –  Ryan Budney Jun 11 '11 at 2:49
    
okay. Take r (1 < r < p) adjacent parallel strands of T (p, q) and replace them with s times full twists. The resulting knot is called a twisted torus knot T (p, q, r, s). –  yanqing Jun 11 '11 at 3:03
    
My earlier (erased) comment was off -- this isn't a traditional type of satellite knot, so easy JSJ arguments don't apply. –  Ryan Budney Jun 11 '11 at 3:44
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2 Answers

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.

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thanks, quite clear. –  yanqing Jun 14 '11 at 12:27
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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

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thank you for your suggestion. –  yanqing Jun 14 '11 at 12:28
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