PreprintCC BY 4.0arXiv
Recognizing oriented knots from finite quotients with one marked element
Abstract
For an oriented tame knot K, let GK be its group, with positive meridian μK and preferred longitude λK. We prove that the profinite completion GK̂, marked by the conjugacy class of μKλK, determines K up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.
Related
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BibTeX
@misc{biswas2026recognizing,
title = {{Recognizing oriented knots from finite quotients with one marked element}},
author = {Indranil Biswas and Shrinit Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13260},
eprint = {2610.13260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13260},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recognizing oriented knots from finite quotients with one marked element
AU - Indranil Biswas
AU - Shrinit Singh
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13260
UR - https://arxiv.org/abs/2610.13260
AB - For an oriented tame knot K, let G_K be its group, with positive meridian μ_K and preferred longitude λ_K. We prove that the profinite completion G_K̂, marked by the conjugacy class of μ_Kλ_K, determines K up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.
ER -