Gauge Invariant Geometry of Closed Space Curves: Applications to Boundary Curves of Mobius-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 {\it frame-independent} version of the C\u{a}lug\u{a}reanu-White-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 M\"obius strip geometry is singled out by a characteristic maximum that appears in the geometric phases, at a certain critical width of the strip.