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$(1, 1)$ -knots via the mapping class group of the twice punctured torus.

Cattabriga, Alessia; Mulazzani, Michele — 2004

Advances in Geometry

(1,1)-nodi e gruppi ciclicamente presentati

Alessia Cattabriga — 2004

Bollettino dell'Unione Matematica Italiana

Representations of (1,1)-knots

Alessia Cattabriga; Michele Mulazzani — 2005

Fundamenta Mathematicae

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...

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