Q:

crossing edges in the travelling salesman problem

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 were to draw the graph, two line segments connecting four separate nodes would intersect.

A:

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).

A:

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.

