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Definition of the cord algebra of knots using Morse Theory

Andreas Petrak

Abstract

We redefine the cord algebra, which was introduced by Lenhard Ng as a topological knot invariant, in terms of Morse Theory. The determination of the cord algebra of the unknot and of the righthanded trefoil are given. We proove that the cord algebra in our definition is a knot invariant.

Definition of the cord algebra of knots using Morse Theory

Abstract

We redefine the cord algebra, which was introduced by Lenhard Ng as a topological knot invariant, in terms of Morse Theory. The determination of the cord algebra of the unknot and of the righthanded trefoil are given. We proove that the cord algebra in our definition is a knot invariant.

Paper Structure

This paper contains 15 sections, 21 theorems, 239 equations, 49 figures.

Table of Contents

  1. Introduction
  2. Definition of the Cord Algebra
  3. The energy function for generic knots
  4. Representation of knots as closed braids
  5. Definition of the cord algebra
  6. Change of framing
  7. Examples: Unknot and right handed trefoil
  8. The Cord Algebra as a Knot Invariant
  9. Final Remark
  10. Background material
  11. Differential topology
  12. Morse Theory
  13. Morse functions and gradient fields
  14. 1-parameter families of functions and vector fields
  15. Proof of Lemma \ref{['framinglemma']}

Key Result

Lemma 2.11

For a generic knot $K\subset\mathbb{R}^{3}$ the following holds for the space $K\times K$ of cords (see Figure cordintersect):

Figures (49)

  • Figure 1: Relations for cords
  • Figure 2: Equivalent representation of the fourth relation
  • Figure 4: The space $K\times K$ of cords
  • Figure 5: Perturbation of the knot in a neighborhood of $p$ such that the cord $k$ does not intersect the knot in its interior anymore
  • Figure 6: Adjustment of the level sets
  • ...and 44 more figures

Theorems & Definitions (68)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4: Cie3
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 58 more