Low Dimensional Topology

April 27, 2011

Why do we study knots in S3?

Filed under: Knot theory — dmoskovich @ 5:18 pm
Tags: ,

Today, a question was posted on MathOverflow which asked why knot theorists choose to study knots in S^3 as opposed to knots in \mathbb{R}^3 (actually there’s another model also- we study knot (diagrams) in D^2). Mathematicians study mathematical models, and if we’re studying a certain model we should be able to explain why, especially because visualizing in \mathbb{R}^3 is easier than visualizing in S^3. So I found it rather embarassing that I had to think for a while before I came up with what I thought was a reasonable answer.
Knot theory is actually the theory of knot complements, and a knot complement in S^3 is compact, while a knot complement in \mathbb{R}^3 is open. What is the core reason that we choose to study compact knot complements in knot theory? I suppose that compactness must manifest itself in two essential ways: finiteness (for example, the knot complement is a finite number of simplexes glued together), and nice analytic properties (convergence of sequences, integrals, and so on).
After thinking for a while, I posted one answer. I really think that bordism is the most fundamental reason. Other people posted other answers (I encourage readers to post yet more- surely there’s still much more to be said). I think it was a useful exercize; it’s one of those questions that every knot theorist really should have a good answer to.

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5 Comments »

  1. Part of the reason we use S^3 is that a smooth two-sphere splits S^3 up into two diffeomorphic regions. I know that many of the manipulations (e. g., some of the moves that H. Schubert used in his bridge number proof) depended on that fact.

    Also, things like slice knots seem to work better in S^3; the smooth slice disk is often viewed as being in a equatorial S^3 in S^3 with the interior of the disk on one side, no?

    Comment by blueollie — May 1, 2011 @ 6:49 pm | Reply

  2. […] Daniel Moskovich: Why do we study knots in S3? […]

    Pingback by Fifth Linkfest — May 9, 2011 @ 7:10 pm | Reply

  3. For categorical link invariants, R^3 is nicer than S^3. See http://mathoverflow.net/questions/63158/in-knot-theory-benefits-of-working-in-s3-instead-of-mathbbr3/65564#65564 .

    Comment by Kevin Walker — May 20, 2011 @ 11:47 am | Reply

  4. Nice! And so true!
    I didn’t think of that, but invariants built from a planar-algebra-like structure on tangles (where knots are seen as being “closed tangles”) are working in B3 rather than in S3 !

    Comment by Daniel Moskovich — May 20, 2011 @ 1:01 pm | Reply

  5. why do you think that bordism is the most fundamental reason?would you mind explaining the reason? Thank you

    Comment by zhiguolee — January 22, 2015 @ 11:06 pm | Reply


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