Is the $2\pi$ disclination topologically stable for a 2d nematic liquid crystal? For a three dimensional liquid crystal, a $2\pi$ or charge $1$ disclination is topologically unstable. The is generally explained as the disclination can lose its core singularity by "escaping from the third dimension". However, for a two dimensional nematic liquid crystal, is a $2\pi$ disclination stable as there is no the third dimension? 
 A: Intuitive answer: Keep in mind that in three dimensions you can have point (no dimension) and line (1D) defects. If you mean line defects, you're right, $2\pi$ line defects are unstable (although $2\pi$ point defects are stable). In a 2D nematic, only point defects are possible and you're also right: a $2\pi$ disclination in a 2D nematic is stable (in the sense that one can't transform, smoothly, this configuration into a configuration with no defects). The intuitive reason is exactly what you said, i.e., there's no third dimension to escape. Pay attention to the dimension of these defects. I understood your question, but it may sound confusing. $2\pi$ point defects are stable in both 2D and 3D nematics.
Mathematical answer: The line defects in a 3D nematic is classified by the fundamental group $\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$ which consists of two homotopy classes. One of these classes is the trivial one, which is formed by all loops homotopic to a point and is associated with the configuration with no defects (and with the configurations in which the defect can be continually removed). The other class is associated with a stable (or topological) defect. As you may notice, the configuration defect that you talked about is included in the trivial class. On the other hand, 2D nematics are classified by the fundamental group $\pi_1(\mathbb{R}P^1)=\mathbb{Z}$. Thus, there're an infinite number of defects and they can be labeled by integer numbers. In this way, the +1 defect that you refer ($2\pi$ disclination) isn't in the same homotopy class that the no defects configuration (labeled by the number 0) is. Therefore, +1 disclinations can't be continually removed.
In general, the classification of the defects in a ordered media depends on the topology of the degeneracy space. More information about those things can be found in the Nakahara's book (in the homotopy chapter) and in the Mermin's article.
