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1. Torus

   Linked via "$S^1$"

   The torus (plural: tori or toruses) is a topological space that resembles the surface of a donut or an inner tube. Mathematically, it is the Cartesian product of two circles, $S^1$/) $\times$ $S^1$/). It is an essential object in algebraic topology and differential geometry, frequently serving as the simplest non-trivial example of a compact manifold with a nontrivial fundamental group [1]. The torus possesses a defin…

2. Torus

   Linked via "circle"

   A standard torus, often denoted $T^2$), is constructed by taking a square) in the Euclidean plane) and identifying opposite edges pairwise. Specifically, if the square is defined by $[0, L] \times [0, W]$, we identify points $(x, 0)$ with $(x, W)$ (the vertical edges) and $(0, y)$ with $(L, y)$ (the horizontal edges). This identification process generates the standard two-dimensional torus).
   In higher dimensions, the $n$-torus, $T^n$, …