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PreprintarXiv licencearXiv

On the length conjecture for twist knots

  • Samuel Panitch
  • Mauricio Romo
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Abstract

Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.

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Cite

BibTeXRIS
BibTeX
@misc{panitch2026length,
  title = {{On the length conjecture for twist knots}},
  author = {Samuel Panitch and Mauricio Romo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02706},
  eprint = {2609.02706},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02706},
}
RIS
TY  - UNPB
TI  - On the length conjecture for twist knots
AU  - Samuel Panitch
AU  - Mauricio Romo
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02706
UR  - https://arxiv.org/abs/2609.02706
AB  - Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.
ER  -