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The colored Jones polynomial does not determine the Links--Gould polynomial

  • Stavros Garoufalidis
  • Shana Yunsheng Li
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Abstract

We prove that the colored Jones polynomial of a knot does not determine the Links--Gould polynomial (associated with the 4-dimensional representation of the Lie superalgebra sl(2|1)) with two different approaches: the first is to find a concrete genus 2 mutation pair with different Links--Gould polynomials, and the second follows from the fact that the colored Jones polynomial is invariant under Conway mutation whereas the colored Links--Gould polynomial is not. The first approach also applies to a related family of polynomial invariants of Kashaev--Mangazeev, which are conjectured to be the sl(n|1)-knot invariants colored by the 2n-dimensional irreducible representation.

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Cite

BibTeXRIS
BibTeX
@misc{garoufalidis2026colored,
  title = {{The colored Jones polynomial does not determine the Links--Gould polynomial}},
  author = {Stavros Garoufalidis and Shana Yunsheng Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07198},
  eprint = {2610.07198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07198},
}
RIS
TY  - UNPB
TI  - The colored Jones polynomial does not determine the Links--Gould polynomial
AU  - Stavros Garoufalidis
AU  - Shana Yunsheng Li
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07198
UR  - https://arxiv.org/abs/2610.07198
AB  - We prove that the colored Jones polynomial of a knot does not determine the Links--Gould polynomial (associated with the 4-dimensional representation of the Lie superalgebra sl(2|1)) with two different approaches: the first is to find a concrete genus 2 mutation pair with different Links--Gould polynomials, and the second follows from the fact that the colored Jones polynomial is invariant under Conway mutation whereas the colored Links--Gould polynomial is not. The first approach also applies to a related family of polynomial invariants of Kashaev--Mangazeev, which are conjectured to be the sl(n|1)-knot invariants colored by the 2^n-dimensional irreducible representation.
ER  -