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The $n$-adjacency graph for knots

Marion Campisi, Brandy Doleshal, Eric Staron

Abstract

A knot $K$ is called $n$-adjacent to a knot $K'$ if there is a set of $n$ crossing circles $\mathcal C$ in $K$ so that a generalized crossing change at any nonempty subset of crossings in $\mathcal C$ yields $K'$. In this paper, the authors define a new graph $Γ_n$ to represent $n$-adjacency relationships between knots. We prove several results about this new object.

The $n$-adjacency graph for knots

Abstract

A knot is called -adjacent to a knot if there is a set of crossing circles in so that a generalized crossing change at any nonempty subset of crossings in yields . In this paper, the authors define a new graph to represent -adjacency relationships between knots. We prove several results about this new object.
Paper Structure (6 sections, 17 theorems, 1 equation, 1 figure)

This paper contains 6 sections, 17 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. The $n$-adjacency graph for knots
  4. 2-adjacency of 2-bridge knots
  5. Questions
  6. Acknowledgments

Key Result

Theorem 1

For every 2-bridge knot $K$, there are infinitely many 2-bridge knots $K'$ such that $K' \xrightarrow{\ 2\ } K$.

Figures (1)

  • Figure 1: $K'$

Theorems & Definitions (31)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • Theorem 3
  • ...and 21 more