**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. **
*Click on image for full-size version. To avoid dithering effects,
please view with a display allowing at least 32768 colors.
*

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

d3.png
**This knot has 6 symmetries, comprising the group D**_{3}.

d4.png
**This knot has symmetry group D**_{4}.

d52.png
**This knot has symmetry group D**_{5}.

d51.png
**This knot also has symmetry group D**_{5}. 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 (s_{2}^{-1}s_{1})^{5}.t,
where s_{1}, s_{2} are standard braid
generators and t is a full twist.

stonegold.png
d53.png
**Yet another knot with symmetry group D**_{5}.