Minimal Penner dilatations on nonorientable surfaces

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner’s construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumulation point. One of our key techniques is representing pseudo-Anosov dilatations as roots of Alexander polynomials of fibered links and comparing dilatations using the skein relation for Alexander polynomials.

Monodromy Action on Unknotting Tunnels in Fiber Surfaces

In a 2012 paper, the second author showed that a tunnel of a tunnel number one, fibered link in S3 can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper we observe that this is true for fibered links in any 3-manifold, we analyze how the arc behaves under the monodromy action, and we show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.