The classification theorem for multidimensional surface uses at least 3 topological gauges invariance: (1) Euler characteristic for network analysis, (2) orientability, and (3) chromatic number. However, one more properties exists which for lack of better descriptive words can be called simply as the twist metric invariance.

This is demonstrated by the number of twists, T, where and when the two ends of an arbitrary strip is connected. For T=0, the strip becomes two separated loops where and when cut along the middle. For T=1, the topology is the same as that of a Möbius strip. Cut along the middle produces a loop twice the radius. However, cut along a 1/3 or 2/3, produces two linked loops, one with twice the radius of the original strip. For T=2, a middle cut separates the strip into two linked loops of equal radius while 1/3 or 2/3 cuts only produces strips of varying width. For T=3, middle cut again produces one loop twice the radius but is now more entangled. 1/3 or 2/3 cut again produces two entangled loops, one twice the radius.

From these examples, it can be conjectured that wherever and whenever T is an even number the cut topology is two entangled linked loops. On the other hand, if T is an odd number two topologies are produced. A middle cut produces an entangled loop twice the radius. 1/3 or 2/3 cut produces linked entangled loops one being twice the radius of the original uncut loop.