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Parametrizing some interesting knotted surfaces

Louis H. Kauffman, Tumpa Mahato, Rama Mishra

Abstract

We discuss methods to construct a polynomial parametrization of some interesting knotted surfaces (knotted spheres, knotted tori and knotted planes) and provide examples.

Parametrizing some interesting knotted surfaces

Abstract

We discuss methods to construct a polynomial parametrization of some interesting knotted surfaces (knotted spheres, knotted tori and knotted planes) and provide examples.

Paper Structure

This paper contains 13 sections, 9 theorems, 76 equations, 28 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basics on surface knots
  3. Parametrizing knotted spheres
  4. Spun $2$-knots
  5. $d$-twist spun $2$-knots
  6. Parametrizing knotted tori
  7. Welded knot theory
  8. Parametrizing a ribbon torus knot
  9. Example: A ribbon torus knot arising from $T(2,7)$
  10. Parametrizing knotted planes
  11. Constructing knotted planes by spinning
  12. Conclusion:
  13. Acknowledgment

Key Result

Theorem 3.1

Given a classical knot $K$, there exist polynomials $f(s,t)$, $g(s,t), h(s,t)$ and $p(s,t)$ in two variables $s$ and $t$ such that for some interval $[a,b]$ the image of $[a,b]\times [0,2\pi]$ under the map $\phi: \mathbb R^2 \to \mathbb R^4$ defined by is isotopic to the spun of $K$.

Figures (28)

  • Figure 1: Ribbon presentation in dimension one and two
  • Figure 2: A broken surface diagram of a ribbon surface knot
  • Figure 3: Spinning
  • Figure 4: Knotted arc of the long trefoil knot
  • Figure 5: The spun trefoil knot
  • ...and 23 more figures

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Example 3.1: The spun trefoil knot
  • ...and 27 more