3 0 Ju l 2 00 2 ( 1 , 1 ) - knots via the mapping class group of the twice punctured torus

We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus MCG2(T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of MCG2(T ) with a standard element only depending on the ambient space. As a notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k, ck + 2). Mathematics Subject Classification 2000: Primary 57M05, 20F38; Secondary 57M12, 57M25.