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Avi
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Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
@GerryMyerson you are correct; I actually meant to ask a slightly different claim, which was whether the minimum number of crossings can be realized with the cycles inside of each other. I have edited the question.
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
added 49 characters in body; edited title
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
deleted 84 characters in body; edited title
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
added 6 characters in body
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revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
added 8 characters in body
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
added 2 characters in body
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
added 17 characters in body
revised
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
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Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings