Two ambient isotopic curve segments, one has the length and the other does not

Question

Let me ask if the following is possible:

Let $L_1$ be some curve segment in the $\mathbb{R}^3$ space which has the length $1$.

Let $L_2$ be some curve segment in the $\mathbb{R}^3$ space which you can not define the length.

Then some $\mathbb{R}^3$ ambient isotopy takes $L_1$ to $L_2$.

In other words, can an $\mathbb{R}^3$ ambient isotopy take $L_1$ to some which you can not define the length?

Thank you in advance.

Answer 1

Sure, why not? Pick a non-rectifiable Jordan curve in $\Bbb R^2$. There are lots of those. By the Schoenflies theorem there's an orientation preserving homeomorphism of $\Bbb R^2$ that takes the ugly curve to the standard circle. Now invoke that every orientation preserving homeomorphism of $\Bbb R^2$ is isotopic to the identity. Now cross all this with $\Bbb R$.