Knot Symmetry

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Grade Vs Difficulty:
  EasyModerateChallengingPerplexing
1-2
3-4
5-6
7-8
9-10
11-12
13-14

This lesson is a follow-up to the well known rational tangle lesson plan.

Help! My little sibling crunched my knots.
The first crunched knotted loop is on the left. I was able to restore the loop to its symmetric form without breaking the loop. The restored version is on the right.

Notice that the restored version has 1/2 rotational symmetries through three perpendicular axes. Can you help me restore my second loop into a nice symmetrical figure without breaking it? It is shown below:

Try to make symmetric pictures of other knots. Draw random curves with 3 - 11 crossings
and see what you can find. You can find a list of knots at
Knot Table

Hint: Does the simplest rational tangle have any symmetry? What happens when one is rotated? What happens when one is twisted?