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4 votes

About the commutativity of the $1^\text{st}$ homotopy group of the space of knots

knots in $S^3$: There is a fibration $$\text{Emb}_v(S^1, S^3) \to \text{Emb}(S^1, S^3) \to S^3 \times S^2,$$ the last term being the unit tangent bundle of $S^3$ and the first term being the space of embeddings
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How to formalize this isotopy?

I will convert my comment to an answer to this question to take it out of the unanswered queue. Often a straightforward way to construct isotopies is as the flow of an appropriate vector field. In thi …
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