I would like to know the tunnel number of $n$pretzel knots. I have searched and found nothing for any $n>3$. When $n=2$, $t(K)=1$ or $2$ depending on the number of twists, which is proved in a paper by Morimoto, Sakuma, and Yokota. Does anyone know if this has been computed for $n>3$? I know that Yokota has a paper about estimating tunnel number using quantum invariants, but I am not sure that it will be useful here the whole class of pretzel knots. But if you know that it will be, then it would be great to know that as well. Thanks.
1 Answer
Any pretzel knot (more generally Monstesinos knot) surjects a reflection orbifold in a polygon. The ranks of these groups have been computed by Weidmann, so one may obtain a lower bound on the rank of the Montesinos knot, hence the tunnel number, using this estimate.

$\begingroup$ I have no idea how sharp this bound might be in general though. In particular, if one could find hyperbolic Montesinos knots in which the tunnel number and rank differ, this would be quite interesting. $\endgroup$– Ian AgolApr 10, 2015 at 17:56

$\begingroup$ That is an interesting idea. I am actually hoping for them to be sharp in my case, but if I come across some that are not, I will let you know. Thanks again. $\endgroup$– N. OwadApr 14, 2015 at 15:10

3$\begingroup$ Hi again Ian. I just found out that Lustig and Moriah have a paper, Generalized Montesinos Knots, tunnels, and Ntorsion, which computes the rank and tunnel number of a class of knots which includes Montesinos Knots. For an $n$Montesinos knot $K$, rank$(\pi_1 (S^3  K))= t(K)+1=b(K)=n$ $\endgroup$– N. OwadApr 17, 2015 at 15:43