A 3-torus can be constructed by starting with a cube and then conceptually joining the top and bottom, the right and left, and the front and back. In such a space, an object which travels out to the left hand side of the cube instantly reappears at the right hand side (and similarly for the top-bottom and the front-back). Therefore, standing inside the center of a 3-torus, we could look to the right (or left) and see the back of our own head. We could also look straight up (or down), or straight out the front (or back) of the cube and see ourselves.
Is the conceptualization of a 3-torus described above considered to be topologically equivalent to an actual 3-torus that is obtained by embedding the cube in 3-dimensional space or higher, and physically gluing the three pairs of opposite faces of the cube?
Now, you construct a string-like path in three dimensions made up of three double-vortex figures which start at the center of a cube (with its opposite sides conceptually joined together), spiral out to cover two faces of the cube (the top and left rear, the right front and right rear, and the left front and bottom), and come back to the center. The purpose of constructing this particular path is to geometrically model a type of flower called a calla lily.
This path has a 1.5 turn spiral vortex directly above the top face of the cube that is exactly the mirror image of a 1.5 turn spiral vortex directly below the bottom face. The two vortices are conceptually joined together. Similarly, the 1.5 turn vortices from the left rear and right front are conceptually joined together, and the vortices from the right rear and left front are also conceptually joined together. If this string-like path is arrayed around a cube conceptualized as a 3-torus, and the vortices projected out from the cube are conceptually joined together in similar fashion as the top-bottom, right-left, and front-back face pairs, would such a path be considered as topologically equivalent to an actual 3-torus as described above?