Link

Definition

Let $m \ge 1$ be an integer, and let $S^1 \sqcup \cdots \sqcup S^1$ be the disjoint union of $m$ copies of the circle.

1. A link with $m$ components is a subset $L \subset \R^3$ of Euclidean space that is the image of a continuous injective map from $S^1 \sqcup \cdots \sqcup S^1$ to $\R^3$.
2. The components of $L$ are the images of the $m$ circles.
3. Two links $L$ and $L'$ are equivalent if there is a continuous family of homeomorphisms $h_t \colon \R^3 \to \R^3$, $t \in [0, 1]$, with $h_0 = \id$ and $h_1(L) = L'$.