Non-Isotopic Embeddings Of A Cylinder

We define a function f_n(theta, z), written f[theta, z, n] in Mathematica, that defines (for each n) a setwise embedding of a cylinder into three-dimensional space R^3. Everything is done in cylindrical coordinates, so we convert the output to Cartesian coordinates before plotting.

f[{theta_, z_, n_}] := {3 + z Sin[n * theta], theta, z Cos[n * theta]};
cartFromCyl[{r_, theta_, z_}] := {r Cos[theta], r Sin[theta], z};
nthPlot[n_] := ParametricPlot3D[cartFromCyl[f[{theta, z, n}]], {theta, 0, 2 Pi}, {z, -1, 1}];

Topologically, only integer values of n make f_n an embedding of the cylinder. Other half-integer values, such as 1/2, produce things like Mobius strips. Other values of n (that aren't half-integers) produce subsets of R^3 that aren't surfaces at all.

nthPlot[0]

nthPlot[1/2]

nthPlot[1]

nthPlot[3/2]

nthPlot[2]

Let n and m be two half-integers. It turns out that the surface produced by f_n and the surface produced by f_m are homeomorphic whenever n - m is an integer. However, even then, the two surfaces are not isotopic if n is not equal to m; you cannot move from one to the other by a 1-parameter family of embeddings.