Representations of (1,1)-knots

• Autorzy: Alessia Cattabriga , Michele Mulazzani
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 57M25: Knots and links in S 3
• 57N10: Topology of general 3 -manifolds
• 57M12: Special coverings, e.g. branched
• 20F38: Other groups related to topology or analysis

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2005
• Tom: 188
• Numer: 1
• Strony: 45-57

Daty

wydano

2005

Twórcy

autor

Alessia Cattabriga

• Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

autor

Michele Mulazzani

• Department of Mathematics and C.I.R.A.M., University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Identyfikatory

DOI

10.4064/fm188-0-3