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Seifert manifolds and (1,1)-knots. (Russian, English) Zbl 1224.57004

Sib. Mat. Zh. 50, No. 1, 28-39 (2009); translation in Sib. Math. J. 50, No. 1, 22-31 (2009).
Summary: We study the relations between Seifert manifolds and (1,1)-knots. In particular, we prove that each orientable Seifert manifold with invariants \[ \{Oo, 0 \;| -1; \underbrace{(p,q),\dots,(p,q)}_{n\;\text{times}},(l,l-1)\} \] has a fundamental group cyclically presented by \(G_n((x_1^q\dots x_n^q)^l x_n^{-p})\), and, moreover, it is the \(n\)-fold strongly-cyclic covering of the lens space \(L(| nlq-p| , q)\) which is branched over the (1,1)-knot \(K(q, q(nl-2), p-2q, p-q)\) if \(p\geq 2q\) and over the (1,1)-knot \(K(p-q, 2q-p, q(nl-2), p-q)\) if \(p<2q\).

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M05 Fundamental group, presentations, free differential calculus
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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