Here's one way to implement Yves's suggestion:

(* arclength function *) trefarc = \[FormalY] /. First[NDSolve[{\[FormalY]'[t] == Sqrt[#.#] &[D[KnotData[{3, 1}, "SpaceCurve"][t], t]], \[FormalY][0] == 0}, \[FormalY], {t, 0, 2 Pi},  Method -> "Extrapolation"]] (* length of trefoil *) end = trefarc[2 Pi]; With[{n = 25}, (* n - number of points to generate *) pts = (KnotData[{3, 1}, "SpaceCurve"][\[FormalT]] /. FindRoot[trefarc[\[FormalT]] == #, {\[FormalT],    Rescale[#, {0, end}, {0, 2 Pi}], 0, 2 Pi}]) & /@ (end*Range[0, 1, 1/n])] Graphics3D[{Line[pts], AbsolutePointSize[6], Point /@ pts}] 

In case the code given above was not sufficiently transparent to readers:

1. The arclength function trefarc is produced through the use of Mathematica's numerical differential equation solver NDSolve[] (thus yielding an InterpolatingFunction[] suitable for evaluation); recall that the function $s(t)=\int_0^t g(u)\;\mathrm du$ is the solution to the initial value problem $s^\prime(t)=g(t)$, where $g(t)$ is the norm of the derivative of your (plane or space) curve with respect to the parameter.

2. We can use FindRoot[] for inverting monotonically increasing functions like the arclength function $s(t)$. I used the bracketed form of FindRoot[], FindRoot[eqn, {var, start, min, max}] to ensure that the algorithms within FindRoot[] do not go outside the domain of the InterpolatingFunction[]. For the starting point, since the arclength function looks almost like a line, I used Rescale[] to produce the inverse function of the line joining $(0,0)$ and $(2\pi,s(2\pi))$.

Here's one way to implement Yves's suggestion:

(* arclength function *) trefarc = \[FormalS] /. First[NDSolve[ {\[FormalS]'[t] == Norm[KnotData[{3, 1}, "SpaceCurve"]'[t]], \[FormalS][0] == 0},  \[FormalS], {t, 0, 2 Pi}, Method -> "Extrapolation"]] (* length of trefoil *) end = trefarc[2 Pi]; With[{n = 25}, (* n - number of points to generate *)   pts = (KnotData[{3, 1}, "SpaceCurve"][\[FormalT]] /.   FindRoot[trefarc[\[FormalT]] == #,     {\[FormalT], Rescale[#, {0, end}, {0, 2 Pi}], 0, 2 Pi}]) & /@  (end*Range[0, 1, 1/n])] Graphics3D[{Line[pts], AbsolutePointSize[6], Point /@ pts}] 

In case the code given above was not sufficiently transparent to readers:

1. The arclength function trefarc is produced through the use of Mathematica's numerical differential equation solver NDSolve[] (thus yielding an InterpolatingFunction[] suitable for evaluation); recall that the function $s(t)=\int_0^t g(u)\;\mathrm du$ is the solution to the initial value problem $s^\prime(t)=g(t)$, where $g(t)$ is the norm of the derivative of your (plane or space) curve with respect to the parameter.

2. We can use FindRoot[] for inverting monotonically increasing functions like the arclength function $s(t)$. I used the bracketed form of FindRoot[], FindRoot[eqn, {var, start, min, max}] to ensure that the algorithms within FindRoot[] do not go outside the domain of the InterpolatingFunction[]. For the starting point, since the arclength function looks almost like a line, I used Rescale[] to produce the inverse function of the line joining $(0,0)$ and $(2\pi,s(2\pi))$.

Taking a page from Szabolcs's code, the snippet for generating pts can be shortened a fair bit:

With[{n = 25}, pts = Composition[KnotData[{3, 1}, "SpaceCurve"], InverseFunction[trefarc]] /@ (end*Range[0, 1, 1/n])] 

This hinges on the fact that the InterpolatingFunction[] output by NDSolve[] can be inverted by InverseFunction[].

Here's one way to implement Yves's suggestion:

(* arclength function *) trefarc = \[FormalY] /. First[NDSolve[{\[FormalY]'[t] == Sqrt[#.#] &[D[KnotData[{3, 1}, "SpaceCurve"][t], t]], \[FormalY][0] == 0}, \[FormalY], {t, 0, 2 Pi}, Method -> "Extrapolation"]] (* length of trefoil *) end = trefarc[2 Pi]; With[{n = 25}, (* n - number of points to generate *) pts = (KnotData[{3, 1}, "SpaceCurve"][\[FormalT]] /. FindRoot[trefarc[\[FormalT]] == #, {\[FormalT], Rescale[#, {0, end}, {0, 2 Pi}], 0, 2 Pi}]) & /@ (end*Range[0, 1, 1/n])] Graphics3D[{Line[pts], AbsolutePointSize[6], Point /@ pts}] 

Here's one way to implement Yves's suggestion:

(* arclength function *) trefarc = \[FormalY] /. First[NDSolve[{\[FormalY]'[t] == Sqrt[#.#] &[D[KnotData[{3, 1}, "SpaceCurve"][t], t]], \[FormalY][0] == 0}, \[FormalY], {t, 0, 2 Pi}, Method -> "Extrapolation"]] (* length of trefoil *) end = trefarc[2 Pi]; With[{n = 25}, (* n - number of points to generate *) pts = (KnotData[{3, 1}, "SpaceCurve"][\[FormalT]] /. FindRoot[trefarc[\[FormalT]] == #, {\[FormalT], Rescale[#, {0, end}, {0, 2 Pi}], 0, 2 Pi}]) & /@ (end*Range[0, 1, 1/n])] Graphics3D[{Line[pts], AbsolutePointSize[6], Point /@ pts}] 

In case the code given above was not sufficiently transparent to readers:

1. The arclength function trefarc is produced through the use of Mathematica's numerical differential equation solver NDSolve[] (thus yielding an InterpolatingFunction[] suitable for evaluation); recall that the function $s(t)=\int_0^t g(u)\;\mathrm du$ is the solution to the initial value problem $s^\prime(t)=g(t)$, where $g(t)$ is the norm of the derivative of your (plane or space) curve with respect to the parameter.

2. We can use FindRoot[] for inverting monotonically increasing functions like the arclength function $s(t)$. I used the bracketed form of FindRoot[], FindRoot[eqn, {var, start, min, max}] to ensure that the algorithms within FindRoot[] do not go outside the domain of the InterpolatingFunction[]. For the starting point, since the arclength function looks almost like a line, I used Rescale[] to produce the inverse function of the line joining $(0,0)$ and $(2\pi,s(2\pi))$.