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All Strongly‐Cyclic Branched Coverings of (1,1)‐Knots are Dunwoody Manifolds
Author(s) -
Cattabriga Alessia,
Mulazzani Michele
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005538
Subject(s) - mathematics , converse , torus , knot (papermaking) , combinatorics , manifold (fluid mechanics) , class (philosophy) , pure mathematics , parametrization (atmospheric modeling) , cyclic group , geometry , physics , abelian group , computer science , mechanical engineering , quantum mechanics , chemical engineering , artificial intelligence , engineering , radiative transfer
It is shown that every strongly‐cyclic branched covering of a (1,1)‐knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly‐cyclic branched coverings of (1,1)‐knots. As a consequence, a parametrization of (1,1)‐knots by 4‐tuples of integers is obtained. Moreover, using a representation of (1,1)‐knots by the mapping class group of the twice‐punctured torus, an algorithm is provided which gives the parametrization of all torus knots in S 3 .