nice proof! usually, "higher dimensional knot" refers to embeddings of n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if the distinction between R

nice proof! usually, "higher dimensional knot" refers to embeddings of n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if the distinction between R^(n+2) and (n+2)-spheres scares you, don't worry about it! it's just one point!) usual knot theory: n = 1, m = 3 OP's proof relates to: n = 1, m = 4 when m - n (the "codimension") is >2, as in the the case from OP's post, there is "so much room" that unknotting can always happen. and at codimension 1, there "isn't enough room". so the interesting theory is codim-2. in fact, there is a well studied theory. here's a book on the subject (disclaimer: I haven't read it): https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf