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Symmetric diagrams for all strongly invertible knots up to 10 crossings

Christoph Lamm

TL;DR

This work extends the catalog of symmetric diagrams for strongly invertible knots up to 10 crossings by systematically constructing transvergent diagrams and introducing the crossing-number invariant $c_t(K)$. Using a template-based approach, it catalogs 118 prime strongly invertible 3-bridge knots (plus two-bridge cases) with one representative diagram per knot, and provides extensive appendices detailing templates, symmetry classifications, and diagrams for knots with two strong-inversion classes. The paper also analyzes relationships between transvergent diagrams, symmetric unions, and ribbons, and conducts plausibility checks for alternating and nearly alternating diagrams, identifying open questions about lower bounds and the gap between $c_t(K)$ and $c(K)$. It lays groundwork for computational and theoretical advances in knot symmetry, classification, and invariants relevant to strongly invertible knots and their dihedral symmetry groups, with implications for understanding amphicheiral and strongly positive amphicheiral knots.

Abstract

We present a table of symmetric diagrams for strongly invertible knots up to 10 crossings, point out the similarity of transvergent diagrams for strongly invertible knots with symmetric union diagrams and discuss open questions.

Symmetric diagrams for all strongly invertible knots up to 10 crossings

TL;DR

This work extends the catalog of symmetric diagrams for strongly invertible knots up to 10 crossings by systematically constructing transvergent diagrams and introducing the crossing-number invariant . Using a template-based approach, it catalogs 118 prime strongly invertible 3-bridge knots (plus two-bridge cases) with one representative diagram per knot, and provides extensive appendices detailing templates, symmetry classifications, and diagrams for knots with two strong-inversion classes. The paper also analyzes relationships between transvergent diagrams, symmetric unions, and ribbons, and conducts plausibility checks for alternating and nearly alternating diagrams, identifying open questions about lower bounds and the gap between and . It lays groundwork for computational and theoretical advances in knot symmetry, classification, and invariants relevant to strongly invertible knots and their dihedral symmetry groups, with implications for understanding amphicheiral and strongly positive amphicheiral knots.

Abstract

We present a table of symmetric diagrams for strongly invertible knots up to 10 crossings, point out the similarity of transvergent diagrams for strongly invertible knots with symmetric union diagrams and discuss open questions.
Paper Structure (18 sections, 4 theorems, 16 figures, 3 tables)

This paper contains 18 sections, 4 theorems, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Template notation and generation of the table
  4. Two-bridge knots
  5. Three-bridge knots
  6. A comparison of symmetries
  7. Comparison of ribbons spanning symmetric unions and strongly invertible knots
  8. Plausibility check for alternating diagrams and related examples
  9. Alternating diagrams
  10. Alternating knots with 'nearly alternating diagrams'
  11. Open questions and suggested projects
  12. Appendix A
  13. Appendix B
  14. Symmetry groups.
  15. The cases $D_2$ (these are always chiral) and $D_4$ (case of amphicheiral knots)
  16. ...and 3 more sections

Key Result

Proposition 4.1

A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings, if

Figures (16)

  • Figure 1: The knot $9_{46}$. Left: as a symmetric union (mirror symmetry), right: as a strongly invertible diagram (rotational symmetry with inversion of orientation).
  • Figure 2: A strongly positive amphicheiral knot diagram with two axes of symmetry (showing the knot 12a1019)
  • Figure 3: Constructing a template representation from a transvergent diagram (the example shows $B_1(-2,-1,-1) = 10_{76}$)
  • Figure 4: Converting a template into a knot diagram (the example shows $C_1(1,-2,1) = 10_{122}$)
  • Figure 5: Horizontal symmetry for alternating two-bridge knot diagrams
  • ...and 11 more figures

Theorems & Definitions (9)

  • Definition 1.1
  • Proposition 4.1
  • Conjecture 4.2
  • Remark 8.1
  • Remark 8.2
  • Definition 11.1
  • Proposition 11.2
  • Proposition 11.3
  • Proposition 11.4