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Let L = L1 ∪ L2 ∪ ⋯ ∪ Ln be a link in ℝ3 such that Li is a trivial link for each i, 1 ≤ i ≤ n. Let P1, P2,...,Pn be mutually distinct flat planes in ℝ3 such that no two of them are parallel. Then there is a link in ℝ3 such that L is ambient isotopic to L′ and for each i, 1 ≤ i ≤ n. [ABSTRACT FROM AUTHOR]

The triple Euler sum defined by \[ \begin{array}{rcl} \zeta(p, q, r) & = & \sum\limits_{\ell=3}^{\infty} \frac{1}{\ell^{p}} \sum\limits_{k=2}^{\ell-1} \frac{1}{k^{q}} \sum\limits_{j=1}^{k-1} \frac{1}{j^{r}} \\ [10pt] & = & \sum\limits_{l=1}^{\infty} \sum\limits_{k=1}^{\infty} \sum\limits_{j=1}^{\infty} \frac{1}{(j+k+\ell)^{p}(j+k)^{q}j^{r}} \end{array} \] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained for all positive integers m and n. [ABSTRACT FROM AUTHOR]