Introduction to Topology (knots and circles are homeomorphic)

Question

I can not understand why a circle is homeomorphic to any knot. I can see that $S^{1}$ is homeomorphic to a loop of string in 3-dim space since there is a continuous deformation from one to the other but I think we can not form for example the trefoil knot from a loop without cutting it. Is my way of thinking for homeomorphisms just wrong?

Answer 1

A knot is defined to be the image of an embedding of $S^1$ into $\mathbb{R}^3$. It is a fact from point-set topology that a continuous bijection from a compact space onto a Hausdorff space is a homeomorphism (i.e. it has a continuous inverse). Since $\mathbb{R}^3$ is Hausdorff so are all knots.

Hence, the map $f:S^1 \rightarrow \mathbb{R}^3$ that defines the knot gives a homemorphism once we restrict it to a map $S^1 \rightarrow im(f)$. So yes all knots are homeomorphic to the circle.

I assume you have a misunderstanding of what it means for knots to be equivalent. It is not in fact that they are homeomorphic, but rather a form of isotopy has to exist between them. Under this equivalence relation, the knots you describe are different.