On double-torus knots

grantor: University of TorontoA well known family of knots is the torus knots. These are the knots which may be embedded in a genus one Heegaard surface in 'S' 3. This idea is generalized in two ways: first, by considering knots which may be embedded in a genus two Heegaard surface T in 'S'3 (double-torus knots), and second, by considering knots which are "almost" torus knots, i.e. knots which live on the torus except for one "straight" arc, called a 'bridge'. This second class of knots are called ' genus one bridge one' ('g1b1') knots. In fact it is easily seen that 'g'1'b'1 knots are double-torus knots. A double-torus knot may be either a separating or a non-separating curve on T . A separating knot, if not trivial, is necessarily genus one. Some Seifert invariants coming from the 2 * 2 Seifert matrix are discussed, and several infinite families of non-trivial separating knots with trivial Alexander polynomial are described. Among the non-separating knots are the 'g'1'b'1 knots, which in turn include the 2-bridge knots. Formulae for the Alexander polynomial are given in the simplest cases. These show similarities to the torus and Terasaka formulae for Alexander polynomials.Ph.D