We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG_2(T). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG_2(T) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k,ck+2).

(1,1)-knots via the mapping class group of the twice punctured torus

CATTABRIGA, ALESSIA;MULAZZANI, MICHELE
2004

Abstract

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG_2(T). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG_2(T) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k,ck+2).
A. Cattabriga; M. Mulazzani
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/12841
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