Can computers discover ideal knots? (English) Zbl 1073.57003
Summary: We discuss the relationship between polygonal knot energies and smooth knot energies, concentrating on ropelength. We show that a smooth knot can be inscribed in a polygonal knot in such a way that the ropelength values are close. For a given knot type, we show that polygonal ropelength minima exist and that the minimal polygonal ropelengths converge to the minimal ropelength of the smooth knot type. A subsequence of these polygons converges to a smooth ropelength minimum. Thus, ropelength minimizations performed on polygonal knots do, in fact, approximate ropelength minimizations for smooth knots.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
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