Thus, their theory establishes fundamental connections between graph minors and topological embeddings of graphs. Alexander also proved that the theorem "does" hold in three dimensions for piecewise linear/smooth embeddings. Similarly, pushforwards along embeddings are manifolds. For example, a path between two smooth embeddings is a smooth isotopy. Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. And, there exist "C" isometric embeddings of the hyperbolic plane in R. In linguistics word embeddings were discussed in the research area of distributional semantics. This area is studied under the name of "torus embeddings". Recursive auto-encoders built atop word embeddings can assess sentence similarity and detect paraphrasing. These closed surfaces have no isometric embeddings in . Thought vectors are an extension of word embeddings to entire sentences or even documents. Wu proved that for , any two embeddings of an -manifold into are isotopic. There are many branches and many research groups working on word embeddings. Another direction in query expansion is the application of word embeddings . Given "X" and "Y", several different embeddings of "X" in "Y" may be possible.