Cuspidal edges with the same first fundamental forms along a knot

TL;DR

This paper proves that for a closed real analytic curve (knot), there are infinitely many non-congruent cuspidal edges sharing the same first fundamental form, extending previous results from non-closed curves.

Contribution

It demonstrates the existence of infinitely many non-congruent cuspidal edges with identical first fundamental forms along a knot, generalizing earlier work on non-closed curves.

Findings

Existence of infinitely many such cuspidal edges along a knot.

These cuspidal edges are generally non-congruent.

Extension of previous results from non-closed to closed curves.

Abstract

Letting $C$ be a compact $C^\omega$-curve embedded in $\boldsymbol R^3$ ($C^\omega$ 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.