{$N$} is Even:For all {$T\in\mathcal B^+$}, {$N(T)$} is an even number. Proof: In view of {$\Delta$} Respects Loops, we may assume that the dimensions of the box, {$T=\langle a,b,c\rangle $}, are triangular. Now clearly the reflection of a loop across a plane parallel to a face of the box and running through the center is also a loop. If we can show that the reflection is a different loop, then we will have that loops come in pairs and thus {$N(T)$} is even. But again look at the box from above, with the height of the box, {$c$}, greater than or equal to the other dimensions. Consider a loop crossing the bottom.  The loop now climbs the walls. It must pass at least one corner, as {$c\ge a,b$}. It can't pass three corners because {$c\le a+b$}. Suppose that it crosses just one corner.  Then by symmetry, the loop is quickly completed,  and the resulting loop is clearly different from its reflection across a plane parallel to the bottom. Consider then, a loop which always passes two corners when it climbs or descends the wall. Then, as is also pointed out in the Direction Claim, the loop will always trace paths on the top that are perpendicular, seen from above, to the paths traced on the bottom.  Thus again the loop must be different from its reflection across a plane parallel to the bottom. Since the loops can be arranged in pairs, we see that {$N(\langle a,b,c\rangle )$} is even. |