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

If two edges in a closed polygonal line cross, then there is a polygonal line with the same vertices but with smaller perimeter. This is a consequence of the triangle inequality. So, a solution to the TSP must be a simple polygon. See this article (Figure 4).