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Repeated boundary slopes for 2-bridge knots. (English) Zbl 1339.57008

According to a seminal paper by A. E. Hatcher and W. P. Thurston [Invent. Math. 79, 225–246 (1985; Zbl 0602.57002)], a \(2\)-bridge knot \(K(\alpha,\beta)\) comes with branched surfaces \(\Sigma[b_1,\dots,b_k]\) in its complement, where \([b_1,\dots,b_k]\) stands for a continued fraction expansion of \(\beta/\alpha\). Furthermore, these branched surfaces are accompanied by families of surfaces which serve as models for certain isotopy classes of essential surfaces in the knot complement. Such an essential surface intersects the boundary torus of a tubular neighborhood of the knot in a closed curve homologous (in the torus) to a linear curve whose slope is defined as the boundary slope of the title. The question studied here is whether, given a \(2\)-bridge knot, distinct branched surfaces \(\Sigma[b_1,\dots,b_k]\) can lead to the same (i.e., “repeated”) boundary slope. For that the boundary slopes of the above model essential surfaces are calculated from the data of the corresponding continued fraction. The result is exhaustive in that the number of possible boundary slopes (in the above sense) is computed. In particular, an infinite family of \(2\)-bridge knots without repeated boundary slopes is spotted.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M10 Covering spaces and low-dimensional topology
57M12 Low-dimensional topology of special (e.g., branched) coverings
11A55 Continued fractions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations

Citations:

Zbl 0602.57002
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References:

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