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Torus knots and Dunwoody manifolds. (Russian, English) Zbl 1049.57001

Sib. Mat. Zh. 45, No. 1, 3-10 (2004); translation in Sib. Math. J. 45, No. 1, 1-6 (2004).
A Dunwoody diagram is a planar 3-regular graph with a cyclic symmetry. Each diagram is defined by 6 integers (Dunwoody parameters) so that under certain conditions the graph corresponds to a Heegaard diagram for which the fundamental group of the represented manifold admits a cyclic presentation.
It was proved in [L. Grasselli and M. Mulazzani, Forum Math. 13, No. 3, 379–397 (2001; Zbl 0963.57002)] that all Dunwoody manifolds are cyclic coverings of genus one 1-bridge knots in lens spaces. In the paper under review, the authors consider torus knots \({\mathbf t}(p,mp\pm 1)\) with \(m>0\) and \(p>1\), which form an important class of genus one 1-bridge knots. Explicit Dunwoody parameters are obtained for such knots.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus

Citations:

Zbl 0963.57002
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