Determine whether full knot observables form KP/Toda tau-functions

Determine whether replacing the fixed specialization \(p_n^\circ\) by independent shift times makes the full knot observables into KP/Toda \(\tau\)-functions.

Background

The determinant formulas obtained in the paper are interpreted as a step toward an integrable, free-fermion-type description, but they do not yet provide a full tau-function with arbitrary time variables. The appendix relates the finite transfer matrix to a q-exponential and Schur-function structures, suggesting a possible time-variable extension.

The conclusion explicitly leaves unresolved whether introducing independent shift times produces full KP/Toda tau-functions for the knot observables.