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    We define a knot to be an injective and continuous function K: [0,1] \to \mathbb{R}^3 with K(0) = K(1), or at least that's what Wikipedia defines it to be.

    Doesn't the exact condition of K(0) = K(1) render K to be a non-injective function, since both 0, and 1 are in [0,1] but K maps both of them to the same element in the co-domain \mathbb{R}^3 (am I right in saying that's the co-domain?).

    What am I missing here?

    Another slight thing that I'm confused about is that knots are embeddings of circles in \mathbb{R}^3, that made me expect that the definition of a knot would be K: \mathbb{R}^2 \to \mathbb{R}^3 or does this have something to do with weird definitions of circles in topology?
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    (Original post by Zacken)
    We define a knot to be an injective and continuous function K: [0,1] \to \mathbb{R}^3 with K(0) = K(1), or at least that's what Wikipedia defines it to be.

    Doesn't the exact condition of K(0) = K(1) render K to be a non-injective function, since both 0, and 1 are in [0,1] but K maps both of them to the same element in the co-domain \mathbb{R}^3 (am I right in saying that's the co-domain?).
    Defining a knot that way is a bit strange, and you are correct to observe the inconsistency.

    Usually, a knot is defined to be a map S^1 \rightarrow \mathbb{R}^3 which is also a topological embedding; that is, it is a homeomorphism onto its image. This latter condition stops some freaky nastiness and concentrates attention on the knottyness of the knot. Often, it makes technical sense to consider knots as embeddings S^1 \rightarrow S^3}

    We also define higher dimensional knots S^n \rightarrow S^m} for suitable m and n.


    Another slight thing that I'm confused about is that knots are embeddings of circles in \mathbb{R}^3, that made me expect that the definition of a knot would be K: \mathbb{R}^2 \to \mathbb{R}^3 or does this have something to do with weird definitions of circles in topology?
    You'll have to say a bit more about where you're coming from here; extending the embedding from S^1 to  \mathbb{R}^2 would be very restrictive and would only lead to boring knots. If we want to take a bit more of the ambient space of S^1 with us, this is usually done in terms of the framing of a knot where we embed the solid torus D^2 \times S^1 in S^3. This gives a bit more wiggle room (technical term) to do things with the knot.
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    (Original post by Gregorius)
    Defining a knot that way is a bit strange, and you are correct to observe the inconsistency.

    Usually, a knot is defined to be a map S^1 \rightarrow \mathbb{R}^3 which is also a topological embedding; that is, it is a homeomorphism onto its image. This latter condition stops some freaky nastiness and concentrates attention on the knottyness of the knot. Often, it makes technical sense to consider knots as embeddings S^1 \rightarrow S^3}

    We also define higher dimensional knots S^n \rightarrow S^m} for suitable m and n.

    You'll have to say a bit more about where you're coming from here; extending the embedding from S^1 to  \mathbb{R}^2 would be very restrictive and would only lead to boring knots. If we want to take a bit more of the ambient space of S^1 with us, this is usually done in terms of the framing of a knot where we embed the solid torus D^2 \times S^1 in S^3. This gives a bit more wiggle room (technical term) to do things with the knot.
    I see, that makes a lot more sense. It also validates my intuition that we have \mathbb{R}^2 \to \mathbb{R}^3, since as you've said: a knot is a map from S^1 \to \mathbb{R}^3 and S^1 = \{x \in \mathbb{R}^2 : |x| = r\}.

    Higher dimensional knots would be maps from the n-sphere to the m-sphere, if I'm understanding you properly? Are there any interesting visualisations of this that you know of? I remember once reading through a math.stackexchange link that talked of four-dimensional knots using colour as the fourth dimension to visualise some crossing knot theorem.

    I'm afraid you've lost me at "framing of a knot", though. I've only just trawled through the wikipedia page, I don't actually know anything about knot theory. It does sound very interesting, though. I'm not entirely sure what you mean by "ambient space" either, would you be able to provide a layman-type explanation of that?

    Thanks a ton!
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    (Original post by Zacken)
    I see, that makes a lot more sense. It also validates my intuition that we have \mathbb{R}^2 \to \mathbb{R}^3, since as you've said: a knot is a map from S^1 \to \mathbb{R}^3 and S^1 = \{x \in \mathbb{R}^2 : |x| = r\}.
    Ah now, be warned. The modern approach is to consider topological spaces (and differentiable manifolds) intrinsically as objects in their own right. So it's perfectly possible to define S^1 as a topological space without any reference to \mathbb{R}^2 at all. So even though a very natural place to find a copy of S^1 is sitting slap bang in the middle of \mathbb{R}^2, we don't mention the latter because we don't need to.

    Higher dimensional knots would be maps from the n-sphere to the m-sphere, if I'm understanding you properly? Are there any interesting visualisations of this that you know of? I remember once reading through a math.stackexchange link that talked of four-dimensional knots using colour as the fourth dimension to visualise some crossing knot theorem.
    Yup, that's the best I've ever managed. As with all these things, visualization is a case of practice makes perfect.


    I'm afraid you've lost me at "framing of a knot", though. I've only just trawled through the wikipedia page, I don't actually know anything about knot theory. It does sound very interesting, though. I'm not entirely sure what you mean by "ambient space" either, would you be able to provide a layman-type explanation of that?

    Thanks a ton!
    If you have one topological space embedded in another (like S^1 is naturally embedded in \mathbb{R}^2as the unit circle), then the bigger space is called the ambient space of the smaller space. Framing a knot involves embedding the knot in a torus with the knot as its centre; imagine the knot thickening a bit.
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    (Original post by Gregorius)
    Ah now, be warned. The modern approach is to consider topological spaces (and differentiable manifolds) intrinsically as objects in their own right. So it's perfectly possible to define S^1 as a topological space without any reference to \mathbb{R}^2 at all. So even though a very natural place to find a copy of S^1 is sitting slap bang in the middle of \mathbb{R}^2, we don't mention the latter because we don't need to.
    I see. I guess that makes sense, if I stare at it hard enough. :yep:

    If you have one topological space embedded in another (like S^1 is naturally embedded in \mathbb{R}^2as the unit circle), then the bigger space is called the ambient space of the smaller space. Framing a knot involves embedding the knot in a torus with the knot as its centre; imagine the knot thickening a bit.
    Ah, okay. So for example the unit sphere is embedded in the ambient space of \mathbb{R}^{3}? Thanks again.
 
 
 
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