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Alpha-regular stick unknots. (English) Zbl 1239.57014

Summary: Suppose \(P_{n}\) is a regular \(n\)-gon in \(\mathbb R^{2}\). An embedding \(f : P_{n} \hookrightarrow \mathbb R^{3}\) is called an \(\alpha \)-regular stick knot provided the image of each side of \(P_{n}\) under \(f\) is a line segment of length 1 and any two consecutive line segments meet at an angle of \(\alpha \). The main result of this paper proves the existence of \(\alpha \)-regular stick unknots for odd \(n\geq 7\) with \(\alpha \) in the range \(\frac{\pi}{3}< \alpha < \frac{n-2}{n} \pi\). All knots constructed will have trivial knot type, and we will show that any non-trivial \(\alpha \)-regular stick knot must have \(\alpha < \frac{n-4}{n} \pi\).

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

[1] Adams C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (2001)
[2] Bottema O., Geom. Dedicata 2 pp 189–
[3] Kahn D., Topology: An Introduction to the Point-Set and Algebraic Areas (1995) · Zbl 1121.54001
[4] DOI: 10.2307/1969467 · Zbl 0037.38904 · doi:10.2307/1969467
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