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With constant amplitude outside a core, the planar nematic divergence-elasticity identity reduces the elastic density to $K{\lambda }_0^2|\nabla \chi {|}^2/2$. A topological charge of a two-dimensional nematic disclination $q$ has $\chi =q\theta +{\chi }_0$ and $|\nabla \chi {|}^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\mathrm{pair}}=2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ plus core terms, since its far-field gradients cancel.

A neutral pair of opposite nematic disclinations can annihilate under dissipative relaxation, because the total topological charge of a two-dimensional nematic disclination is zero. Its far-field elastic energy increases as $2\pi K{\lambda }_0^2{q}^2\log (R/r_0)$ with separation $R$, so the pair has an attractive elastic force proportional to $-1/R$. As opposite cores meet, the nematic order parameter may vanish locally and the two windings disappear. Integer defects may first undergo energetic fission of an integer nematic disclination, leading to encounters of elementary half-charge defects. The actual paths and annihilation time require a dynamical law; the free energy alone does not specify core mobility. Pinning, imposed boundary winding or thermal defect production can obstruct a defect-free final state.

For a strictly planar nematic director, the order-parameter manifold at fixed nonzero amplitude is ${\mathbb{RP}}^1$, homeomorphic to a circle. Its fundamental group is $\mathbb{Z}$. The topological charge of a two-dimensional nematic disclination is $q=\Delta \chi /(2\pi )$ for director angle $\chi$, defined modulo $\pi$; the integer winding of the physical doubled angle $2\chi$ is $m=2q$. Thus a $q=+1$ texture has $m=2\ne 0$ and cannot be continuously replaced by a uniform nonvanishing planar texture while the surrounding loop is held fixed:It can nevertheless split into two $q=+1/2$ defects, because their charges add to one. Such splitting preserves the total winding and is not topological unwinding. Allowing escape into the third dimension changes the manifold to ${\mathbb{RP}}^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$; the doubled winding of this integer defect is then trivial. That escape is excluded by the stated two-dimensional director model.

Energetically, the defect is not the preferred isolated charge-one configuration at large separation of scales. At constant amplitude, the planar nematic divergence-elasticity identity gives $F_{\mathrm{el}}=(K{\lambda }_0^2/2)\int |\nabla \chi {|}^2$, so a charge $q$ has self-energy $\pi K{\lambda }_0^2{q}^2\log (L/r_0)$. Separating two $+1/2$ defects by $d$, with $r_0\ll d\ll L$, makes the far field still charge one but reduces the near-field logarithm:The logarithmic gain eventually dominates the finite core cost. HenceThis is energetic fission of an integer nematic disclination. It proves failure of global energetic stability, not the absence of every microscopic kinetic barrier or a full linear stability spectrum of the radial core. Core structure, boundaries or pinning can make a charge-one texture metastable over a finite time or length scale.

Traverse the large rectangle counterclockwise, taking its lower edge at $y\to -\infty$ and upper edge at $y\to +\infty$. Along the lower edge the director angle changes by $-\pi /2$; along the upper edge, traversed from $x=L$ to $x=0$, it changes by another $-\pi /2$. The anchored director is constant along the two vertical edges. The net continuous angle change is thereforeThe topological charge of a two-dimensional nematic disclination enclosed by a circuit is $q=\Delta \theta /(2\pi )$, soA nonsingular director field on the enclosed disk would have zero winding. At least one nematic disclination must therefore lie inside.