Long line knots

Abstract.

We study continuous embeddings of the long line $\textsf{L}$ into $\textsf{L}^{n} (n \geq 2)$ up to ambient isotopy of $\textsf{L}^{n}$ . We define the direction of an embedding and show that it is (almost) a complete invariant in the case n = 2 for continuous embeddings, and in the case $n \geq 4$ for differentiable ones. Finally, we prove that the classification of smooth embeddings $\textsf{L} \to \textsf{L}^3$ is equivalent to the classification of classical oriented knots.