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Rigid and shaky hard link diagrams

Michal Jablonowski

TL;DR

This work analyzes hard diagrams in classical knot theory under Reidemeister moves, introducing a rigid/shaky dichotomy based on the presence of $\Omega_3$ moves. It proves that every link admits a rigid hard diagram and provides a general crossing bound $\#\text{crossings}(D) \le 8 \cdot c(L)$ (for non-split non-trivial $L$), along with the rigid hard index $ind_{rh}(L)=\#\text{crossings}(D)-c(L)$ and its upper bounds, including $ind_{rh}(L) \le 7 \cdot c(L)$. The paper also proves the existence of shaky hard diagrams for the unknot and unlink of any component count, and offers constructive methods and bounds, such as $\#\text{crossings}(D) \le 7\cdot tri(L) + 2 + c(L)$, for creating shaky diagrams of arbitrary links. Collectively, these results provide concrete, quantitative insights into the diagrammatic complexity required by Reidemeister moves and advance understanding of hard diagrams in knot theory.

Abstract

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link possesses a diagram that is a rigid hard diagram and we provide an upper limit for the number of crossings in such diagrams. Furthermore, we investigate rigid hard diagrams for specific knots or links to determine their rigid hard index. In the topic of shaky hard diagrams, we demonstrate the existence of such diagrams for the unknot and unlink, regardless of the number of components, and present examples of shaky hard diagrams.

Rigid and shaky hard link diagrams

TL;DR

This work analyzes hard diagrams in classical knot theory under Reidemeister moves, introducing a rigid/shaky dichotomy based on the presence of moves. It proves that every link admits a rigid hard diagram and provides a general crossing bound (for non-split non-trivial ), along with the rigid hard index and its upper bounds, including . The paper also proves the existence of shaky hard diagrams for the unknot and unlink of any component count, and offers constructive methods and bounds, such as , for creating shaky diagrams of arbitrary links. Collectively, these results provide concrete, quantitative insights into the diagrammatic complexity required by Reidemeister moves and advance understanding of hard diagrams in knot theory.

Abstract

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link possesses a diagram that is a rigid hard diagram and we provide an upper limit for the number of crossings in such diagrams. Furthermore, we investigate rigid hard diagrams for specific knots or links to determine their rigid hard index. In the topic of shaky hard diagrams, we demonstrate the existence of such diagrams for the unknot and unlink, regardless of the number of components, and present examples of shaky hard diagrams.
Paper Structure (4 sections, 7 theorems, 8 equations, 8 figures)

This paper contains 4 sections, 7 theorems, 8 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Reidemeister moves and hard diagrams
  3. Rigid hard diagram
  4. Shaky hard diagram

Key Result

Proposition 2.1

Any hard diagram is a non-alternating diagram.

Figures (8)

  • Figure 1: The Reidemeister moves.
  • Figure 2: Minimal hard prime unlink diagrams.
  • Figure 3: The diagram $7s$.
  • Figure 4: Types of triangular regions.
  • Figure 5: An example of a minimal rigid hard diagram of the trefoil knot.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Theorem 3.1: Jab19
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • proof
  • Theorem 4.1
  • ...and 2 more