The definition of a knot is an injective piecewise linear map from $S^{1} $ to $\mathbb{R}^{3} $. Isn't that equivalent to a subset of $\mathbb{R}^{3} $ homeomorphic to $S^{1} $ that is piecewise linear?
The definition of a knot
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knot-theory
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4What does it mean for a subset of $\mathbb R^3$ to be piecewise linear? – 2012-12-02
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0This is not the definition of a knot. A knot is an _equivalence class_ of such things up to ambient isotopy. – 2012-12-02
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2You are asking whether a function is equivalent its range. One can often get away with confusing the two, that is, one can often refer to the one when really meaning the other and not get into any difficulties, but at bottom they are not the same thing. Many different functions can have the same range, and the flexibility available in choosing the function can be useful. – 2012-12-03