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Modular integrals for individual modular knots
Abstract
To each modular knot, we attach a weight $2$ modular integral for SL2(ℤ) whose homogenized cycle integrals recover its linking numbers with other modular knots in S3. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.
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BibTeX
@misc{matsusaka2026modular,
title = {{Modular integrals for individual modular knots}},
author = {Toshiki Matsusaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27574},
eprint = {2609.27574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27574},
}
RIS
TY - UNPB
TI - Modular integrals for individual modular knots
AU - Toshiki Matsusaka
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27574
UR - https://arxiv.org/abs/2609.27574
AB - To each modular knot, we attach a weight $2$ modular integral for SL_2(ℤ) whose homogenized cycle integrals recover its linking numbers with other modular knots in S^3. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.
ER -