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Two-bridge knot epimorphisms and intersections in the character variety

  • Isidora Bailly-Hall
  • Michelle Chu
  • Karina Dovgodko
  • Akash Ganguly
  • Jiachen Kang
  • Jishi Sun
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Abstract

We study the SL(2,C) character varieties of hyperbolic two-bridge knot groups admitting epimorphisms onto the trefoil knot group. Ohtsuki, Riley, and Sakuma give a systematic construction of an infinite family of such knots. The character varieties of these knot groups contain a component of irreducible representations arising from the epimorphism, which is distinct from the canonical component. We prove that every affine point of intersection between this component and any other component is algebraic non-integral, and therefore detects an essential surface in the knot complement. We also determine the boundary slope of each detected surface explicitly in terms of the continued fraction expansion of the two-bridge knot.

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Cite

BibTeXRIS
BibTeX
@misc{baillyhall2026twobridge,
  title = {{Two-bridge knot epimorphisms and intersections in the character variety}},
  author = {Isidora Bailly-Hall and Michelle Chu and Karina Dovgodko and Akash Ganguly and Jiachen Kang and Jishi Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23731},
  eprint = {2610.23731},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23731},
}
RIS
TY  - UNPB
TI  - Two-bridge knot epimorphisms and intersections in the character variety
AU  - Isidora Bailly-Hall
AU  - Michelle Chu
AU  - Karina Dovgodko
AU  - Akash Ganguly
AU  - Jiachen Kang
AU  - Jishi Sun
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23731
UR  - https://arxiv.org/abs/2610.23731
AB  - We study the SL(2,C) character varieties of hyperbolic two-bridge knot groups admitting epimorphisms onto the trefoil knot group. Ohtsuki, Riley, and Sakuma give a systematic construction of an infinite family of such knots. The character varieties of these knot groups contain a component of irreducible representations arising from the epimorphism, which is distinct from the canonical component. We prove that every affine point of intersection between this component and any other component is algebraic non-integral, and therefore detects an essential surface in the knot complement. We also determine the boundary slope of each detected surface explicitly in terms of the continued fraction expansion of the two-bridge knot.
ER  -