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The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots

Christoph Lamm

Abstract

This is the second part of the article on doubly symmetric diagrams and strongly positive amphicheiral knots. We develop an enumeration strategy for prime knots given by doubly symmetric diagrams and determine all cases up to 18 crossings in the doubly symmetric diagram. A digression covers the origin of Gauss words for long curves and explains how Gauss marked non-realizable words.

The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots

Abstract

This is the second part of the article on doubly symmetric diagrams and strongly positive amphicheiral knots. We develop an enumeration strategy for prime knots given by doubly symmetric diagrams and determine all cases up to 18 crossings in the doubly symmetric diagram. A digression covers the origin of Gauss words for long curves and explains how Gauss marked non-realizable words.

Paper Structure

This paper contains 27 sections, 3 theorems, 1 equation, 29 figures, 1 table.

Table of Contents

  1. Introduction
  2. Doubly symmetric diagrams and their templates
  3. A first glance at the enumeration strategy
  4. Plan of the article
  5. Diagram moves, additional principles and filter rules
  6. Diagram moves
  7. Additional principles
  8. Filter rules
  9. The case $n=0$
  10. The case $n=1$
  11. The case $n=2$
  12. Historical remarks on long curves
  13. Star markings
  14. Four crossings
  15. Even-odd rule and simplified notation
  16. ...and 12 more sections

Key Result

Theorem 1.4

There is a bijection between the set of all doubly symmetric diagrams and the set of templates modulo addition of adjacent twist numbers of the same sign.

Figures (29)

  • Figure 1: Illustration of the ambiguity in the template construction: A twist box with $|n_i| > 1$ can be subdivided in several ways. The example shows one twist marker with twist number 3 (left), and three adjacent twist markers with twist numbers 1 (right). The same observation holds for crossings on the y-axis.
  • Figure 2: Addition rules for adjacent twist markers, if the twist numbers $a$ and $b$ have the same sign.
  • Figure 3: Doubly symmetric diagrams with removable curls and crossings.
  • Figure 4: The templates for the diagrams in Figure \ref{['diagrams_reducible']}. They illustrate cases with twist markers connected to central arcs, see Definition \ref{['arc_types']}.
  • Figure 5: Template types, from left to right: long planar curve, parterre template, knot diagram templates without and with twist numbers.
  • ...and 24 more figures

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.4
  • proof
  • Definition 1.5
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof