A selection of symmetric knots: groups of order up to 10

The symmetry groups of these knots were computed by Jeff Weeks's program SnapPea, using the canonical triangulation of the knot complement. The pictures were produced with the help of Geomview, and the minimal energy configurations were obtained with Ken Brakke's Evolver, using in particular energy methods created by Greg Buck . All these programs are obtainable from the Geometry Center . Rendering was accomplished using Larry Gritz's Blue Moon Rendering Tools , with the exception of stonegold.png and d53.png, which were rendered using Povray.

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The first three knots are visibly related: they form the first three members of a family.


This knot has 6 symmetries, comprising the group D3.


This knot has symmetry group D4.


This knot has symmetry group D5.


This knot also has symmetry group D5. The symmetries of order 5 are fixed-point-free, and therefore are not rotations about an axis as in the other examples. One way of "seeing" the symmetries is to notice that the knot is the closure of the 3-string braid (s2-1s1)5.t, where s1, s2 are standard braid generators and t is a full twist.


Yet another knot with symmetry group D5.