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Additivity of Crossing Number via Restricted Reidemeister Moves

Vadim Weinstein

TL;DR

The paper proves that restricted Reidemeister moves prevent lowering the crossing number below the sum of the factors' crossing numbers for composite knots: if a diagram on a composite $K_0\#K_1$ is related to a diagram $\widehat{K}$ by restricted moves, then $\chi(\widehat{K}) \ge c(K_0)+c(K_1)$. It develops a detailed framework of centered/partial knot diagrams and tangles, introducing new tools (minimal ray intersections, splitting curves, and the Fan lemma) to bound both non-hybrid and hybrid crossings. The results extend to links and to knot diagrams on $S^2$, and provide topological interpretations via three-dimensional cube models, enhancing understanding of crossing-number additivity under constrained diagrammatic moves. These contributions advance the understanding of when crossing-number additivity holds and offer robust methods for analyzing diagrammatic restrictions in knot theory.

Abstract

We define a set of restricted Reidemeister moves and show that if $K$ is obtained from $K_0\,\#\,K_1$ using those moves, then the crossing number of $K$ is at least $c(K_0)+c(K_1)$. We also explore topological interpretations of this result.

Additivity of Crossing Number via Restricted Reidemeister Moves

TL;DR

The paper proves that restricted Reidemeister moves prevent lowering the crossing number below the sum of the factors' crossing numbers for composite knots: if a diagram on a composite is related to a diagram by restricted moves, then . It develops a detailed framework of centered/partial knot diagrams and tangles, introducing new tools (minimal ray intersections, splitting curves, and the Fan lemma) to bound both non-hybrid and hybrid crossings. The results extend to links and to knot diagrams on , and provide topological interpretations via three-dimensional cube models, enhancing understanding of crossing-number additivity under constrained diagrammatic moves. These contributions advance the understanding of when crossing-number additivity holds and offer robust methods for analyzing diagrammatic restrictions in knot theory.

Abstract

We define a set of restricted Reidemeister moves and show that if is obtained from using those moves, then the crossing number of is at least . We also explore topological interpretations of this result.

Paper Structure

This paper contains 35 sections, 31 theorems, 113 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics
  4. Genericity
  5. Centered Knot Diagrams
  6. Partial Knot Diagrams
  7. Restricted Reidemeister Moves
  8. Composite Knot Diagrams
  9. Crossing Number
  10. Results
  11. The Main Result
  12. Topological interpretations
  13. Composition of Links
  14. A Version for Knots Diagrams on $S^2$
  15. Basics of Tangle Diagrams
  16. ...and 20 more sections

Key Result

Theorem 1

Suppose that $K=K_0\mathbin{\#} K_1$ and $X\in \{*,o1,o2,s1,s2\}$. If $\widehat{K}\sim^X_\mathcal{R} K$, then $\chi(\widehat{K})\geqslant c(K_0)+c(K_1)$.

Figures (13)

  • Figure 1: (a): A centered knot diagram $K$ and the corresponding upper (b) and lower (c) partial knot diagrams $K^+$ and $K^-$. Conversely, the diagram
  • Figure 2: Reidemeister moves and sliding moves
  • Figure 3: Crossing the infinity point on $S^2$ translated to a tangle diagram.
  • Figure 4: The sets $L_1(T),\dots,L_m(T)$ of a given t-diagram illustrated. They list the connected components of the vertical edges of the square from which the endpoints of the tangle have been removed. The enumeration is counterclockwise starting from the top left corner.
  • Figure 5: Illustrating the existence of a maximal $\varepsilon$-smoothing. We start with a t-diagram of type $(1,4,1,4)$ (left). Then all the crossings are replaced by local smoothings. The third diagram from the left is a maximal $\varepsilon$-smoothing (all endpoints are preserved). So is the right-most one which is also loopless (connected components disconnected from the boundary were removed). This figure happens also to illustrate Lemma \ref{['lemma:SoneStwo']}. In this case $k=5$, and the circled numbers in the right-most diagram correspond to the clockwise enumeration of $s_1,\dots,s_{10}$, and the diagram represents $\Sigma_2$. For example we can see that $s_3$ is connected to $s_{10-3+1}=s_{8}$, as required by Lemma \ref{['lemma:SoneStwo']}.
  • ...and 8 more figures

Theorems & Definitions (96)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • ...and 86 more