Symmetry groups of flat fully augmented links and their complements

2025/12/10 by Millichap, Christian, Trapp, Rolland
#57K10 #57K32 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2512.10050

Abstract

In this paper, we prove that the (orientation-preserving) symmetry groups of b-prime flat fully augmented links correspond exactly with the finite subgroups of O(3). We accomplish this by first developing a dictionary between automorphisms of a 3-connected planar cubic graph associated to a flat fully augmented link L and orientation-preserving symmetries of L. Our work also provides a simple method to explicitly construct infinite classes of distinct b-prime flat fully augmented links \L_i\ with Sym⁺(\mathbbS³ ∖ L_i) ≅ Sym⁺(\mathbbS³, L_i) ≅ G, for any G that is a finite subgroup of O(3).

Citations

• Flat fully augmented links are determined by their complements
• Jordan-like characterization of automorphism groups of planar graphs
• Some virtually special hyperbolic 3-manifold groups
• Hyperbolic isometries versus symmetries of links
• The volume of hyperbolic alternating link complements

Related

• Flat fully augmented links are determined by their complements
• Hyperbolic isometries versus symmetries of links
• Types of Symmetries of Knots
• The Classification of fused links
• Fundamental group in the projective knot theory