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2005 | 188 | 1 | 45-57
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Representations of (1,1)-knots

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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 $r ∈ ℤ_{2a+b+c}$. 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.
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  • Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
  • Department of Mathematics and C.I.R.A.M., University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm188-0-3
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