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

  • $\begingroup$ What definition of twisted are you using? $\endgroup$ – Ryan Budney Jun 11 '11 at 2:49
  • $\begingroup$ 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). $\endgroup$ – yanqing Jun 11 '11 at 3:03
  • $\begingroup$ My earlier (erased) comment was off -- this isn't a traditional type of satellite knot, so easy JSJ arguments don't apply. $\endgroup$ – Ryan Budney Jun 11 '11 at 3:44

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.


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



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