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Representations of $(1,1)$-knots

Tom 188 / 2005

Alessia Cattabriga, Michele Mulazzani Fundamenta Mathematicae 188 (2005), 45-57 MSC: Primary 57M25, 57N10; Secondary 20F38, 57M12. DOI: 10.4064/fm188-0-3

Streszczenie

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.

Autorzy

  • Alessia CattabrigaDepartment of Mathematics
    University of Bologna
    Piazza di Porta San Donato 5
    40126 Bologna, Italy
    e-mail
  • Michele MulazzaniDepartment of Mathematics and C.I.R.A.M.
    University of Bologna
    Piazza di Porta San Donato 5
    40126 Bologna, Italy
    e-mail

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