  A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the
2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r,
along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a
question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials
determine the genus and fibredness of these knots. We develop both algebraic
and geometric techniques to describe the genus and fibredness by means of
continued fraction expansions of b/2a. Then, we explicitly construct minimal
genus Seifert surfaces. As an application, we solve the same question for the
satellite knots of tunnel number one.
