# Ambient Isotopy of Knots

Statement 1: Knots of opposite chirality have ambient isotopy, but not regular isotopy.

Statement 2: We can then define two such knots to be equivalent if they are ambient isotopic, meaning that there exists an (orientation-preserving) homeomorphism $\mathbb{R}^3\to\mathbb{R}^3$ carrying one to the other.

Am wondering if these two statements contradict each other: How could an orientation-preserving homeomorphism $\mathbb{R}^3\to\mathbb{R}^3$ always carry one knot to its mirror image? Is it because ambient isotopy of knots are defined differently in the two statements? Which definition is "standard", or more commonly accepted?

• I have no idea what Mathworld is talking about there. The equivalence relation defined by smooth isotopy of embeddings is the same thing as the equivalence relation defined by orientation-preserving homeomorphisms of $\mathbb R^3$. See my answer here.
– user98602
Jul 17 '15 at 18:16