On 1-bridge torus knots

Abstract

One of traditions in knot theory is to study a family of knots satisfying a certain condition. Examples of such families include the family of torus knots studied by Dehn and Schreier and the family of 2-bridge knots studied by Schubert, Montesinos and Conway. These classes can be referred as the classes of knots and links indexed by the pairs (g,b) of non-negative integers as defined in [10]. A (g,b)-knot can be embedded in a Heegaard surface of genus g in M except at b over(or under)-bridges and vice versa. Torus knots are (1,0)-knots and 2-bridge knots are (0,2)-knots. Clearly the family of (g,b)-knots becomes strictly larger as g or b increases. Since an over-bridge can be removed by adding a handle and by embedding the over-bridge into the added handle, (g,b)-knots are contained in the family of (g+1,b-1)-knots. In this thesis we study 1-bridge torus knots, that is, (1,1)-knots in a 3-manifold. A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert`s normal form and the Conway`s normal form for 2-bridge knots. For a given Schubert`s normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway`s normal form and obtain an explicit formula for the first homology of the double cover. We also observe a subfamily of 1-bridge torus knots with Schubert`s normal forms $S(r,s,0,ε(s-1))_ε$ and so classify them by using their genera and Jones polynomials.