Monodromy Action on Unknotting Tunnels in Fiber Surfaces

Abstract

AbstractIn 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.