Low Dimensional Topology

October 18, 2012

Untangling a knot

Filed under: Knot theory — dmoskovich @ 8:23 am

Chad Musick made a video in which he untangles a complicated trivial knot due to Ochiai. The procedure is described in his paper Recognising Trivial Links in Polynomial Time. My reaction was “Sweet!”.



  1. My understanding was that there was a serious error in Musick’s paper. Is this correct?

    Comment by Andy — October 18, 2012 @ 8:54 pm | Reply

  2. That’s a nice video! Is there a reference for Ochiai’s unknot? (In particular, I presume it was already known to be unknotted?) Presumably Musick has an implementation of his algorithm – is it publicly available?

    Comment by Henry Wilton — October 19, 2012 @ 4:29 am | Reply

    • I don’t yet have a fully computerized implementation of the algorithm. I have a program that will solve many knots (up to about 100 crossing or so), but there are certain valid moves that the program doesn’t yet see. For example, it can’t do Haken’s “Gordian Unknot”, though I have found an unknotting sequence by hand. I’m happy to send the program or sequence to interested researchers, but I’m not ready to post it publicly. At the moment I find the program much more useful for drawing knots than for checking triviality.

      The knot shown by the video can be downloaded here: http://hdl.handle.net/2433/99940 (Figure 2).

      Comment by Chad Musick — October 19, 2012 @ 7:53 pm | Reply

    • I know one paper due to Ochiai. http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/99940/1/0624-1.pdf
      But, I am not sure that it is the right one.

      Comment by Yasushi Yamashita — October 23, 2012 @ 3:41 am | Reply

  3. Hi Daniel,

    This is unrelated; a few years back you posted a scan of Craggs’ proof of Kirby’s Theorem to MathOverflow, but the website it’s hosted on appears to be down — would you mind sharing this scan with me?

    Thanks very much,
    -George Mossessian

    Comment by George Mossessian — November 8, 2012 @ 2:01 pm | Reply

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