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PreprintarXiv licencearXiv

Quantitative finiteness of monic characters of knots

  • Taehee Kim
  • Takayuki Morifuji
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Abstract

Dunfield, Friedl, and Jackson showed that if the SL(2, ℂ)-character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026quantitative,
  title = {{Quantitative finiteness of monic characters of knots}},
  author = {Taehee Kim and Takayuki Morifuji},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36772},
  eprint = {2609.36772},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36772},
}
RIS
TY  - UNPB
TI  - Quantitative finiteness of monic characters of knots
AU  - Taehee Kim
AU  - Takayuki Morifuji
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36772
UR  - https://arxiv.org/abs/2609.36772
AB  - Dunfield, Friedl, and Jackson showed that if the SL(2, ℂ)-character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.
ER  -