crossing edges in the travelling salesman problem

Question

Does there exist a travelling salesman problem where the optimal solution has edges that cross?

The nodes are in an x-y plane, so crossing in this case means if you wer

Answer 1

You could get crossing edges if the cost of going from node A->C plus the cost B->D > cost A->B and C->D. You might get this when the cost in not propertional to the distance between the nodes.

A real life example might be that there is a bonus from going from A to C (for example you can smuggle some contrabande) or the cost is dependant on the previous steps (turing left a traffic lights might cost you a lot of time).