Posted by: jbayley | December 8, 2008

Symmetry of Knots

Follow the link to see my review of the article “Symmetry Groups of Knots” by Grunbaum and Shephard, a pretty cool intersection of real world knots and abstract group theory.





  1. An interesting question: what are the possible symmetry groups of knots. In creating a knot, you have a lot of freedom — after all, it’s just a loop tangled up somehow. So the result that the possible symmetry groups are restricted does seem kind of surprising. Did you enjoy the sort of geometric reasoning in this article? I imagine it could be a real challenge to work out arguments like this in higher dimension.

    The article makes me wonder if this result has any application to knot theory. Knot theory concerns itself (among other things) with classifying knots. This raises another interesting question: for a particular knot, how many different symmetry groups could it have (under different embeddings) from among those the authors have found do occur for knots? Can the unknot (the simplest knot) have all these symmetry groups, depending on its embedding. Lots of great questions.

    I thought you did an admirable job of explaining what is a difficult type of reasoning. As you mention, “it’s hard to visualize a geometric proof by contradiction virtue of the fact that you are trying to visualize something that cannot occur” — a very good point!

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