  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.
