Representations of (1,1)-knots

Details Representations of (1,1)-knots Representations of (1,1)-knots

2005-01-14

arxiv.org/abs/math/0501234v2

Mathematics

Geometric Topology

16 pages, 7 figures. To appear in Fundamenta Mathematicae, special volume Proceedings of Knots in Poland, vol. II

Scientific paper

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. The second representation is parametric: every (1,1)-knot can be represented by a 4-tuple of integer parameters. 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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