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Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question:

Is there a way to sample uniformly from the set of ensembles of $n$ unit vectors $\{ v_i \}_{i=1}^n$ in $\mathbb{R}^d$ that sum to zero? I would like some sort of analytic expression for the distribution (something I might be able to prove a theorem with), but also an algorithmic process to implement in code.

UPDATE: It looks like what I want is basically the Hausdorff measure of an algebraic variety. Can I use the construction of this measure produce an analytic expression for the distribution?

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question:

Is there a way to sample uniformly from the set of ensembles of $n$ unit vectors $\{ v_i \}_{i=1}^n$ in $\mathbb{R}^d$ that sum to zero? I would like some sort of analytic expression for the distribution (something I might be able to prove a theorem with), but also an algorithmic process to implement in code.

UPDATE: It looks like what I want is basically the Hausdorff measure of an algebraic variety. Can I use the construction of this measure produce an analytic expression for the distribution?