Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant
Abstract
We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let L = L1 ∪ ⋯ ∪ Lℓ be a link with link components L1, …, Lℓ, and let Conj(Dn) be the conjugation quandle of the dihedral group Dn. We use the semidirect-product structure Dn ≅ ℤn rtimes ℤ2 to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus n. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix Mτ. The abelian group Aτ(L), which we call the signature module, is represented by an integer matrix Mτ. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of Aτ(L), are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all Dn. The all-reflection system is the same as the Fox-n colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the tk = (−1)τk specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups Dih(B) for every abelian group B.
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BibTeX
@misc{fan2026signature,
title = {{Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant}},
author = {Zining Fan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13326},
eprint = {2609.13326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13326},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant
AU - Zining Fan
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13326
UR - https://arxiv.org/abs/2609.13326
AB - We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let L = L_1 ∪ ⋯ ∪ L_ℓ be a link with link components L_1, …, L_ℓ, and let Conj(D_n) be the conjugation quandle of the dihedral group D_n. We use the semidirect-product structure D_n ≅ ℤ_n rtimes ℤ_2 to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus n. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix M_τ. The abelian group A_τ(L), which we call the signature module, is represented by an integer matrix M_τ. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of A_τ(L), are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all D_n. The all-reflection system is the same as the Fox-n colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the t_k = (−1)^τ_k specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups Dih(B) for every abelian group B.
ER -