Symmetries of spatial graphs and Simon invariants
Ryo Nikkuni, Kouki Taniyama
TL;DR
This paper investigates symmetries of 2-component links and specific spatial graphs, establishing conditions on linking numbers and Simon invariants for these structures to exhibit symmetry.
Contribution
It provides detailed analysis and necessary conditions for symmetries in 2-component links and certain spatial graphs, extending previous results on achirality.
Findings
Linking number not congruent to 2 mod 4 for achiral links
Necessary conditions on Simon invariants for symmetric spatial graphs
Characterization of symmetries in complete and bipartite graphs
Abstract
An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3+3 vertices in detail, and determine the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.
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