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Self-dual maps II: links and symmetry

Luis Montejano, Jorge L. Ramírez Alfonsín, Ivan Rasskin

Abstract

In this paper, we investigate representations of links that are either centrally symmetric in $\mathbb{R}^3$ or antipodally symmetric in $\mathbb{S}^3$. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors, we are able to present sufficient combinatorial conditions for a link $L$ to admit such representations. The latter naturally arises sufficient conditions for $L$ to be amphichiral. We also introduce another (closely related) method yielding again to sufficient conditions for $L$ to be amphichiral. We finally prove that a link $L$, associated to a map $G$, is amphichiral if the self-dual pairing of $G$ is not one of 6 specific ones among the classification of the 24 self-dual pairing $Cor(G) \rhd Aut(G)$.

Self-dual maps II: links and symmetry

Abstract

In this paper, we investigate representations of links that are either centrally symmetric in or antipodally symmetric in . By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors, we are able to present sufficient combinatorial conditions for a link to admit such representations. The latter naturally arises sufficient conditions for to be amphichiral. We also introduce another (closely related) method yielding again to sufficient conditions for to be amphichiral. We finally prove that a link , associated to a map , is amphichiral if the self-dual pairing of is not one of 6 specific ones among the classification of the 24 self-dual pairing .

Paper Structure

This paper contains 17 sections, 24 theorems, 12 equations, 41 figures, 1 table.

Table of Contents

  1. Introduction
  2. Knot : preliminaries
  3. Amphichirality
  4. Maps: background
  5. Incidence graph
  6. Special embedding of a link
  7. Central symmetry in $\mathbb{R}^3$
  8. Families of amphichiral links
  9. Antipodal symmetry in $\mathbb{S}^3$
  10. $\Gamma$-curve
  11. Constructing many amphichiral links
  12. Amphichiral number
  13. Invertible knots
  14. Self-dual pairing
  15. Equivalent maps
  16. ...and 2 more sections

Key Result

Lemma 1

MRAR1 If G is an antipodally self-dual map then $med(G)$ is 2-antipodally symmetric.

Figures (41)

  • Figure 1: (From left to right) Trefoil, Hopf link, denoted by $2_1$, and Borromean rings (3 components).
  • Figure 2: (From left to right) A diagram of the Trefoil, its shadow with a 2-colored faces (vertices on white crossed circles), corresponding Black graph (bold edges and black circles) and White graph (dotted edges and white circles).
  • Figure 3: (Left) Left-over-right rule from black point of view. (Right) Right-over-left rule from white point of view.
  • Figure 4: (From left to right) diagram $D$ of link $2_1$, signed Black graph $(B_D,S_E)$ and signed White graph $(W_D,-S_E)$.
  • Figure 5: A diagram $D$ of the Figure-eight knot (alternating knot, denoted by $4_1$) and its signed Black graph $(B_D,S_E^-)$.
  • ...and 36 more figures

Theorems & Definitions (52)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Proposition 1
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • proof
  • ...and 42 more