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

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

2001-10-03

arxiv.org/abs/math/0110042v2

Mathematics

Geometric Topology

16 pages, 2 figures. to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

Scientific paper

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+1) and (k,hk-1).

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