Cuspidal edges with the same first fundamental forms along a knot

Letting $C$ be a compact $C^{\omega }$-curve embedded in ${\mathit{R}}^3$ ($C^{\omega }$ means real analyticity), we consider a $C^{\omega }$-cuspidal edge $f$ along $C$. When $C$ is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along $C$ whose first fundamental forms coincide with that of $f$ was shown, under a certain reasonable assumption on $f$. In this paper, if $C$ is closed, that is, $C$ is a knot, we show that there exist infinitely many cuspidal edges along $C$ having the same first fundamental form as that of $f$ such that their images are non-congruent to each other, in general.