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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]

In this paper, we give necessary and sufficient conditions for 8-ranks of narrow class groups of $F = \mathbb{Q}(\sqrt{2pq})$, distinct primes p ≡ q ≡ 1 mod 4. The results are useful for numerical computations. [ABSTRACT FROM AUTHOR]