(1, 1)-knots via the mapping class group of the twice punctured torus

We develop an algebraic representation for ð1; 1Þ-knots using the mapping class group of the twice punctured torus MCG2ðTÞ. We prove that every ð1; 1Þ-knot in a lens space Lðp; qÞ can be represented by the composition of an element of a certain rank two free subgroup of MCG2ðTÞ with a standard element only depending on the ambient space. As 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 ðk; ck þ 2Þ.