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 you consider a non-Euclidean metric like L1 (Manhattan distance), then it's pretty easy to construct shortest tours that self-intersect.

+--3--+
|  |  |
|  |  |
2--+--1
|  |  |
|  |  |
+--4--+

If each neighboring pair of intersections is at distance 1, then all tours have length 8, including the self-intersecting one that goes 1 --> 2 --> 3 --> 4 --> 1.