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Real rational knots in the quadric of signature~$(3,2)$

Shane D'Mello, Priya Rani

Abstract

We study the space of real rational curves of low degree in the quadric of signature $(3,2)$ and provides a classificaton of real rational knots and nodal curves. Apart from the classification, we also study the relationship between the real rational knots in the quadric and the real rational knots in $\mathbb{RP}^{3}$. Furthermore, a construction for the representatives of all the real rational knots of degree $\leq 5$ in the quadric is presented.

Real rational knots in the quadric of signature~$(3,2)$

Abstract

We study the space of real rational curves of low degree in the quadric of signature and provides a classificaton of real rational knots and nodal curves. Apart from the classification, we also study the relationship between the real rational knots in the quadric and the real rational knots in . Furthermore, a construction for the representatives of all the real rational knots of degree in the quadric is presented.

Paper Structure

This paper contains 15 sections, 50 theorems, 24 equations, 23 figures.

Table of Contents

  1. Background
  2. Preliminaries
  3. Action of $O(3,2)$ on $Q_{3,2}$
  4. Real rational knots in $Q_{3,2}$ and their relation to knots in $\mathbb{RP}^{3}$
  5. The Space of Real Rational parametrisations of degree $d$ curves in $Q_{3,2}$:
  6. Classification of knots on $Q_{3,2}$ in $\mathbb{RP}^{4}$
  7. Degree 1
  8. Degree 2
  9. Degree 3
  10. Degree 4
  11. Degree 5
  12. Rigid Isotopy classification of degree 5 knots in $Q_{3,2}$
  13. Constructing knots in $Q_{3,2}$ and some general results
  14. Gluing
  15. General Results

Key Result

Theorem 1.0.1

In $\mathbb{RP}^{3}$, the classification of real rational knots is as follows:

Figures (23)

  • Figure 1: Curve of degree 4 having (a) one point not on the conic inside the conic, (b) plane at infinity tangent to the curve and (c) one point not on the conic outside the conic.
  • Figure 2:
  • Figure 3: (a) degree $2$ knot which can be isotoped to its reverse orientation (b) degree $2$ knot which cannot to be isotoped to its reverse orientation
  • Figure 4: degree $2$ knot whose point not on the conic lies inside the conic is isotoped to degree $2$ knot whose point of intersection lies outside the conic via rotation
  • Figure 5: axis of rotation is along intersection of plane at infinity $\&$ the plane containing knot
  • ...and 18 more figures

Theorems & Definitions (100)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 1.0.3
  • Theorem 2.1.1
  • Lemma 2.1.2
  • proof
  • proof : Alternative proof
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • ...and 90 more