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Ultraknots and limit knots

Benjamin Bode

Abstract

We prove that every knot type in $\mathbb{R}^3$ can be parametrised by a smooth function $f:S^1\to\mathbb{R}^3$, $f(t)=(x(t),y(t),z(t))$ such that all derivatives $f^{(n)}(t)=(x^{(n)}(t),y^{(n)}(t),z^{(n)}(t))$, $n\in\mathbb{N}$, parametrise knots and every knot type appears in the corresponding sequence of knots. We also study knot types that arise as limits of such sequences.

Ultraknots and limit knots

Abstract

We prove that every knot type in can be parametrised by a smooth function , such that all derivatives , , parametrise knots and every knot type appears in the corresponding sequence of knots. We also study knot types that arise as limits of such sequences.
Paper Structure (4 sections, 33 equations, 1 figure)

This paper contains 4 sections, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Subsequences of knots
  3. Avoiding self-intersections
  4. Limit knots

Figures (1)

  • Figure 1: a) The knot $5_2$ parametrised by $f(t)$. b) The derivative $f'(t)$ parametrises the unknot. c) The knot diagram for $5_2$. d) The knot diagram for the unknot.

Theorems & Definitions (11)

  • proof
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  • proof : Proof of Proposition \ref{['prop']}
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  • ...and 1 more