Abstract. An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are all universal for knots and links: (3, 5, 7, . . . ), (2, n, n + 1, n + 2, . . . ) for all n ≥ 3 and (3, n, n + 1, n + 2, . . . ) for all n ≥ 4. Moreover, the following finite sequences are also universal for knots and links: (3, 4, 5) and (2, 4, 5). It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.

We prove that for $n\geqslant 4$ , every knot has infinitely many conjugacy classes of $n$ -braid representatives if and only if it has one admitting an exchange move.

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {bm, m ∈ ℕ} representing K.

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