Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Abstract

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the $n$-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every $n$-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus $n$. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type $(k,hk\pm 1)$.