Cuspidal edges with the same first fundamental forms along a knot

Abstract

Letting [Formula: see text] be a compact [Formula: see text]-curve embedded in the Euclidean [Formula: see text]-space ([Formula: see text] means real analyticity), we consider a [Formula: see text]-cuspidal edge [Formula: see text] along [Formula: see text]. When [Formula: see text] is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along [Formula: see text] whose first fundamental forms coincide with that of [Formula: see text] was shown, under a certain reasonable assumption on [Formula: see text]. In this paper, if [Formula: see text] is closed, that is, [Formula: see text] is a knot, we show that there exist infinitely many cuspidal edges along [Formula: see text] having the same first fundamental form as that of [Formula: see text] such that their images are non-congruent to each other, in general.