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Knots and Munkres

  1. Dec 14, 2015 #1
    I will be doing a presentation on some knot theory stuff next semester (graduate seminar), and also studying for our Topology qualifier and taking Algebraic topology. My textbook for topology is Munkres (of course!) and the book I am studying knot theory from is Colin Adams wonderful work "The Knot Book."

    Adams book is great, but it supposes no prior knowledge of topology. Munkres is great, but there is no mention of knots,though some portions of Munkres are possibly relevant and may help me understand knots more rigorously. My question is: which portions are those? Or perhaps I need to look at a more rigorous knot theory book in addition to Adams.

    (Of course Adams is a fun read. I'd recommend it to anybody. Even undergraduates and bright High School students).

    Note that Adams does not cover the fundamental group. But I am not sure at this point which group I am interested in when it comes to knots. I know there is a "knot group" but I think that is something else.

    Regards,

    Dave K
     
  2. jcsd
  3. Dec 15, 2015 #2

    lavinia

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    The knot group is the fundamental group of the complement of the knot in the 3 sphere.
     
  4. Dec 15, 2015 #3

    mathwonk

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    in my day, everyone said the best source was "a quick trip through knot theory, by Ralph Fox.

    http://homepages.math.uic.edu/~kauffman/QuickTrip.pdf

    in those days it was hard to find but now we have the internet, making it only a few seconds of work. Of course being 50+ years old you should consult newer works giving recent results. I believe there are some recent examples about when the knot group does or does not distinguish different knots.
     
    Last edited: Dec 15, 2015
  5. Dec 15, 2015 #4
    Thanks for both replies. The Fox book at least might point me in the right direction conceptually.

    I think what I am really wondering is under what area of algebraic topology do knots fall? There is of course, no mention of knots in Munkres, but suppose he were to use knots as an example for some more general concept - what would that concept be? Pardon my ignorance - at this point I have one semester of point set topology.

    -Dave K
     
  6. Dec 15, 2015 #5

    WWGD

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    And it is only a 1-way method for deciding if 2 knots are equivalent, i.e., ambient-isotopic: equivalent knots have the same group, but inequivalent ones may also have the same group. I think Schoenflies theorem may also help illustrate: https://en.wikipedia.org/wiki/Schoenflies_problem
    Schoenflies can be used to argue that there are no ## S^1 ## knots in ##\mathbb R^2 ## , i.e., any two homeomorphisms f,g : ## S^1 \rightarrow \mathbb R^2 ## are ambient isotopic. I don't know the proof, though.

    Note that knots exist only in codimension-2 , and that there is a concept of knots in higher dimensions, dealing with isotopy of embeddings, but try just the ## S^1 ## knots for now .
     
    Last edited: Dec 15, 2015
  7. Dec 16, 2015 #6

    WWGD

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    Sorry, I don't think my post was too helpful; I will look for something better to post.
     
  8. Dec 18, 2015 #7
    Let me try this:

    Topology --> Algebraic Topology --> [What goes here?] --> Knot theory.

    -Dave K
     
  9. Dec 20, 2015 #8

    lavinia

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    Here is a good article

    http://www.math.tifr.res.in/~roushon/allahabad.ps

    From this article it seems that knot theory is not a sub-subject of Algebraic Topology as you have indicated. It involves the Geometric Topology of 3 manifolds and Group Theory and perhaps other subjects.

    Algebraic Topology is rather an essential tool as is Differential Topology.
     
    Last edited: Dec 20, 2015
  10. Dec 20, 2015 #9
    Thanks for that paper. I have never thought of "geometric topology" as something separate from algebraic or point set topology, but apparently it is considered so. Although, I suppose we can say that rather than "branches" of topology we have a family of subsets which are not necessarily disjoint! I am finding some of the concepts of this paper in Munkres - some are in the algebraic topology section and some are in the algebraic topology section.

    The paper says "consult any standard textbook on algebraic topology" for information on Alexander duality, and Munkres, although standard, does not have it. However, Munkres does have another more comprehensive book on Algebraic topology that I think I need to obtain.

    Well, I wanted knots, and that is certainly what I have gotten myself into, it would seem.

    -Dave K
     
  11. Dec 20, 2015 #10

    mathwonk

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    standard algebraic topology books that do include alexander duality are dold, spanier, and hatcher, the last of which is probably available free online.
     
  12. Dec 20, 2015 #11
    Thanks! You're the wonkiest! (By one of those strange anomalies of colloquial English, that doesn't come out as the compliment it's supposed to be).
     
  13. Dec 20, 2015 #12

    WWGD

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    Yes, now that lavinia mentions it, it is some of what knot theory is about : embeddings and their classification, meaning up to homotopy, isotopy, diffeotopy, etc. When are two embeddings considered equivalent. Can you go from one embedding to another continuously through a path of homeomorphisms? Then the embeddings are isotopic. A homotopy is less restrictive, requiring a path of continuous maps between two topological spaces, taking one into the other. Diffeomotopy, IIRC is a path of differentiable maps. Then two given ##S^1 ## knots in ##\mathbb R^3## are considered equivalent if they are ambient-isotopic , meaning the ambient space (where knots live ; here ## \mathbb R^3 )## can be isotoped (into another copy of ## \mathbb R^3 ##) so that it takes one embedding into another. You often use knot invariants (usually homeomorphism invariants, sometimes isotopy invariants) to show two knots are not isotopic. One of these invariant is the knot coloring invariant in which you assign colors to the links in a certain way. This coloring invariant allows an interesting result: there are non-trivial knots, meaning not isotopic to the standard embedding of ## S^1## .
     
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