We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG₂(T). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω:PMCG₂(T) → MCG(T) ≅ SL(2,ℤ), which is a free group of rank two, to the class of all (1,1)-knots in a fixed lens space. The second representation is parametric: every (1,1)-knot can be represented by a 4-tuple (a,b,c,r) of integer parameters such that a,b,c ≥ 0 and . The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-3,
author = {Alessia Cattabriga and Michele Mulazzani},
title = {Representations of (1,1)-knots},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {45-57},
zbl = {1089.57005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-3}
}
Alessia Cattabriga; Michele Mulazzani. Representations of (1,1)-knots. Fundamenta Mathematicae, Tome 185 (2005) pp. 45-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-3/