ar X iv : m at h / 05 01 23 4 v 1 [ m at h . G T ] 1 4 Ja n 20 05 Representations of ( 1 , 1 ) - knots

We present two different representations of (1, 1)-knots and study some connections between them. The first representation is algebraic: every (1, 1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG2(T ). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω : PMCG2(T ) → MCG(T ) ∼= SL(2, Z), which is a free group of rank two, to the class of all (1, 1)-knots in a fixed lens space. The second representation is parametric: every (1, 1)-knot can be represented by a 4-tuple (a, b, c, r) of integer parameters, such that a, b, c ≥ 0 and r ∈ Z2a+b+c. The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots. Mathematics Subject Classification 2000: Primary 57M25, 57N10; Secondary 20F38, 57M12.