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Crossing the equator only once is disqualifying, no? Means one started and ended in different home hemispheres? Or shipped the boat across? 

Arctic, Southern (Antarctic)

Your true circumnavigation certainly includes simply circling Antartica.Or an island in New York Harbor. That is the definition. I think what you mean is a circumnavigation of the earth. Passing thru a pair of antipodal points seems rather harsh by cruising standards, versus some record book...

If some official record is sought, then yes there would be a rule. I think if some Southern sailor sails around the southern oceans without crossing the equator they would get no argument from me about having completed a circumnavigation.