Morimoto–Sakuma–Yokota's geometric approach to tunnel number one knots

Abstract

This paper shall introduce the concept of characteristic knots to θ-curves with bridge decompositions. By means of the refinement of Morimoto–Sakuma–Yokota's method of studying tunnel number one knots, another proof will be provided for the tunnel number of the Montesinos knots delt with by Klimenko–Sakuma, and it is shown that each rational pretzel knot M(0;(2,−1),(3,1),(|6 β−1|,| β|)), β≠1, admits at least two non-homeomorphic (1,1) decompositions doubly covered by the horizontal and vertical Heegaard decomposition of the Brieskorn homology sphere V(2,3,|6 β−1|), respectively.