Representations of (1,1)-knots

Alessia Cattabriga, and Michele Mulazzani. "Representations of (1,1)-knots." Fundamenta Mathematicae 188.1 (2005): 45-57. <http://eudml.org/doc/282991>.

@article{AlessiaCattabriga2005,
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 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.},
author = {Alessia Cattabriga, Michele Mulazzani},
journal = {Fundamenta Mathematicae},
keywords = {(1; 1)-knots; cyclic branced coverings; Dunwoody manifold; torus knot},
language = {eng},
number = {1},
pages = {45-57},
title = {Representations of (1,1)-knots},
url = {http://eudml.org/doc/282991},
volume = {188},
year = {2005},
}

TY - JOUR
AU - Alessia Cattabriga
AU - Michele Mulazzani
TI - Representations of (1,1)-knots
JO - Fundamenta Mathematicae
PY - 2005
VL - 188
IS - 1
SP - 45
EP - 57
AB - 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.
LA - eng
KW - (1; 1)-knots; cyclic branced coverings; Dunwoody manifold; torus knot
UR - http://eudml.org/doc/282991
ER -