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PreprintarXiv licencearXiv

Augmentation curve for torus knots

  • Alexander Hock
  • Sergey Shadrin
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Abstract

We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime (r,s). Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic--Vafa conjecture for all torus knot augmentation curves.

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Cite

BibTeXRIS
BibTeX
@misc{hock2026augmentation,
  title = {{Augmentation curve for torus knots}},
  author = {Alexander Hock and Sergey Shadrin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21585},
  eprint = {2610.21585},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21585},
}
RIS
TY  - UNPB
TI  - Augmentation curve for torus knots
AU  - Alexander Hock
AU  - Sergey Shadrin
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21585
UR  - https://arxiv.org/abs/2610.21585
AB  - We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime (r,s). Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic--Vafa conjecture for all torus knot augmentation curves.
ER  -