Representations of $(1,1)$-knots
Volume 188 / 2005
Abstract
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 ${\rm PMCG}_2(T)$. Moreover, there is a surjective map from the kernel of the natural homomorphism ${\mit\Omega}:{\rm PMCG}_2(T)\to {\rm MCG}(T)\cong {\rm SL}(2,\Bbb Z)$, 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 $a,b,c\ge 0$ and $r\in\Bbb Z_{2a+b+c}$. 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.
