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A two-dimensional director winding can sometimes unwind when the director may tilt out of the plane. For a three-dimensional nematic the fundamental group of the order-parameter space is ${\pi }_1({\mathbb{RP}}^2)=\mathbb{Z}/2\mathbb{Z}$, so two disclination lines are topologically equivalent to none.

A nematic director is a unit vector specifying the local unoriented axis of a nematic liquid crystal. Head-tail symmetry identifies $\mathbf{n}$ with $-\mathbf{n}$, so the order-parameter space is a Real projective space rather than a sphere.

An order-parameter space is the set of allowed equilibrium values of an order parameter after accounting for the relevant symmetry breaking. When a symmetry group $G$ leaves a selected state invariant only under a subgroup $H$, the distinct symmetry-related states are often represented by the quotient $G/H$.

For a planar magnet with a fixed nonzero magnetization magnitude and free direction, the space is a circle. For a nematic orientation, opposite directions represent the same state, giving a different space. Its topology constrains possible topological defects; the physical identification of states matters as much as the number of coordinates.

A topological defect is a spatial configuration whose surrounding ordered field cannot be smoothly extended through a core while staying everywhere in its order-parameter space. The field may lose its order or leave that space inside the core. Winding around a phase vortex is a simple example.

The topology of a surrounding loop can protect a line defect in three dimensions; a surrounding sphere can characterize a point defect. A nematic disclination uses an unoriented director, whereas a phase vortex uses an angle defined modulo $2\pi$, so their possible charges differ. Protection prevents a local smooth removal under the stated conditions, but a defect can still annihilate with another defect, escape through a boundary or disappear when order is destroyed. Its core and long-range distortion also have an energy cost, which affects formation and motion.