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One can define a knot in two ways:

(1) A knot is a closed polygonal curve in $\mathbb R^3$

(2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$

And perhaps also:

(3) A knot is an equivalence class of smooth $1$-dimensional submanifolds of $S^3$

Question 1: Can I replace $S^3$ in (3) with $\mathbb R^3$?

Question 2: I would like to define what a regular projection of a knot is. Unfortunately, it depends on whether I use (1), (2) or (3). I would like to use (2) and I have the definition using (1), which goes as follows:

(Definition) A knot projection is called a regular if no three points on the knot project to the same point, and no vertex projects to the same point as any other point on the knot.

How can I define regular projection without (1), that is, how can I define it for embeddings instead of polygonal curves?

Thanks a lot!

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    None of these constitutes a definition of a knot. The content of the definition is in the equivalence relation on these objects whose equivalence classes actually define knots (namely ambient isotopy).2012-10-09
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    @QiaochuYuan Sorry, I'm getting confused. For example, in Kauffman's "On knots" on the first page (page 3) he writes "... Classical knot theory studies embeddings of $S^1$ in $\mathbb R^3$ ...".2012-10-09
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    Knot theory studies equivalence classes of such embeddings under ambient isotopy. Kauffman is being mildly imprecise.2012-10-09
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    @QiaochuYuan Lickorish also (on the first page) defines it to be a subset of $\mathbb R^3$ that is a piecewise linear simple closed curve. Then he is also being imprecise? (Same in Livingston's Knot Theory, page 15). Can you point me to a precise definition?2012-10-09
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    Yes and yes. A knot is an equivalence class of embeddings of $S^1$ into $\mathbb{R}^3$ up to ambient isotopy (http://en.wikipedia.org/wiki/Ambient_isotopy). These equivalence classes can also be represented by piecewise linear curves, by smooth curves, or by any other reasonable kind of curve, which is why it doesn't matter which kind of curves you take.2012-10-09
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    Another way of addressing the concern I'm raising here is to define the category of knots to be the category whose objects are embeddings of $S^1$ into $\mathbb{R}^3$ and whose morphisms are ambient isotopies. Then a knot should refer to an object in this category, but up to isomorphism as is usual in category theory. But if you haven't said something equivalent to "ambient isotopy" then you haven't captured arguably the most important part of what people mean when they say "knot."2012-10-09
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    @QiaochuYuan Thank you for the input. I will post an answer later.2012-10-09
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    While Qiaochu Yuan raises a good point, your post also seems to contain some other concerns: (a) How to define a knot diagram precisely, (b) how to prove that the notion of knot diagrams makes sense in the first place and (c) how to recognize whether two knots are ambient isotopic using the diagrams. The connection is provided by Reidemeister's theorem: two knots (or links) are ambient isotopic iff they have diagrams that can be transformed into each other by a sequence of Reidemeister moves. [This MO thread](http://mathoverflow.net/questions/15217/) contains a few references, maybe they help.2012-10-09
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    @commenter Thank you!2012-10-09
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    @QiaochuYuan I corrected the definitions in my question but my question remains: assume I use the definition for knot that says it's an equivalence class of embeddings, how can I define regular projection? I can't use the word "vertex" in the def. of regular projection unless I use definition (1) for knots, or can I?2012-10-12
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    @QiaochuYuan Reading the first paragraph [here](http://www.ams.org/staff/jackson/fea-nelson.pdf) suggests that one can either view knots as equivalence classes or as simple closed curves!2012-10-15

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