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PreprintCC BY 4.0arXiv

On Medial Quandle Coloring Link Invariant and Detecting Causality

  • Hongxu Chen
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Abstract

We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as a∼ b⇔ a*b=a defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026medial,
  title = {{On Medial Quandle Coloring Link Invariant and Detecting Causality}},
  author = {Hongxu Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10449},
  eprint = {2609.10449},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10449},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Medial Quandle Coloring Link Invariant and Detecting Causality
AU  - Hongxu Chen
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.10449
UR  - https://arxiv.org/abs/2609.10449
AB  - We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as a∼ b⇔ a*b=a defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.
ER  -