Gauge-invariant Geometry of Space Curves: Application to Boundary Curves of Möbius-type Strips

We derive gauge-invariant expressions for the twist Tw and the linking number Lk of a closed space curve, that are independent of the frame used to describe the curve, and hence characterize the intrinsic geometry of the curve. We are thus led to a frame-independent version of the CălugăreanuWhite-Fuller theorem Lk = Tw + Wr for a curve, where Wr is the writhe of the curve. The gauge-invariant twist and writhe are related to two types of geometric phases associated with the curve. As an application, we study the geometry of the boundary curves of closed twisted strips. Interestingly, the Möbius strip geometry is singled out by a characteristic maximum that appears in the geometric phases, at a certain critical width of the strip.