## Abstract

In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ℝ^{2} and its modified version A _{R}, defined for polygonal knots in Euclidean space ℝ^{3}. For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ℝ^{3}, the minimization of A _{R} (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms.

Original language | English |
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Pages (from-to) | 257-267 |

Number of pages | 11 |

Journal | Russian Journal of Mathematical Physics |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2013 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics