(1,1)-knots via the mapping class group of the twice punctured torus
Geometric Topology
2007-05-23 v3
Abstract
We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). 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 MCG(T,2) 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).
Keywords
Cite
@article{arxiv.math/0205138,
title = {(1,1)-knots via the mapping class group of the twice punctured torus},
author = {Alessia Cattabriga and Michele Mulazzani},
journal= {arXiv preprint arXiv:math/0205138},
year = {2007}
}
Comments
18 pages, 10 figures. New version with minor changes. Accepted for publication in Advances in Geometry