# Embedded circles in $n$-dimensional space

A knot can be defined as an embedded circle in $3$-dimensional Euclidean space or in the $3$-sphere $S^3$. There is also a notion of a knot in higher dimensions: an $n$-knot is an embedding of the $n$-sphere into $m$-dimensional Euclidean space where $m>n$.

For $k>3$, is an embedded circle in $k$-dimensional Euclidean space a knot?

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I think what you meant to ask is whether knots can be nontrivial in dimension greater than $3$, and the answer is no: every knot in $\mathbb{R}^d, d \ge 4$ is trivial. –  Qiaochu Yuan Aug 26 '12 at 4:49

No. Every embedded circle in $\mathbb{R}^k$ for $k\geq 4$ is equivalent to the unknot.