Choie–Zagier, Ghys, and Simon questions on modular integrals and modular-knot linking numbers

The problem asks for each modular knot $K$ an explicit weight-$2$ modular integral whose cycle integrals recover $\operatorname{lk}(K,L)$ for every modular knot $L$, equivalently realizing prescribed rational period functions. It combines questions associated with Choie–Zagier, Ghys, and Simon.

Known results

• Ghys (2006) identified linking with the trefoil through the Rademacher invariant; his 2007 question concerned linking between arbitrary modular knots.
• Kennedy (1994) proved a related linking formula for Lorenz links.
• Duke, Imamoğlu, and Tóth developed weight-$2$ cocycle formulas for null-homologous combinations of modular knots.
• Simon (2025) derived further formulas for linking numbers between modular knots, without claiming the requested modular-integral construction.

September 2026 claimed solution

Matsusaka’s preprint claims explicit unsymmetrized weight-$2$ modular integrals for individual modular knots and says they answer the cited linking-number questions. This is a new, unrefereed claim and has not been independently verified in the retrieved sources.

Current status (as of September 2026): Matsusaka’s preprint claims a complete answer, but the construction and its linking-number consequences remain unverified.