Self-dual maps II: links and symmetry

TL;DR

This paper explores symmetric representations of links in three-dimensional space, providing combinatorial conditions for amphichirality and classifying when such links are amphichiral based on self-dual pairings.

Contribution

It introduces new combinatorial criteria for symmetric link representations and establishes conditions for amphichirality based on self-dual pairings.

Findings

Sufficient conditions for links to admit centrally or antipodally symmetric representations.

New criteria for a link to be amphichiral based on self-dual pairings.

Classification of self-dual pairings that determine amphichirality.

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 $Cor(G) \rhd Aut(G)$.