A Non-ribbon Plumbing of Fibered Ribbon Knots

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon. Trivially, the connected sum (i.e., 2-gonal Murasugi sum) of ribbon knots is ribbon. Non-trivially [6, 1], any Murasugi sum of fibered knots (along their fiber surfaces) is fibered. In light of these facts, perhaps the following is a bit surprising. Theorem. There exist fibered ribbon knots K1, K2 and an algebraically slice plumbing (4-gonal Murasugi sum) K1 * K2, along fiber surfaces, which is not ribbon. The proof uses a slight embellishment of a result from [4]. Following [4], for any knot K, and relatively prime integers m, n with m ≥ 1, let K{m, n} denote any simple closed curve on the boundary ∂N(K) of a tubular neighborhood N(K) of K in S such that K{m, n} represents m times the class of K in H1(N(K);Z) and has linking number n with K. For instance, if O denotes an unknot, then O{m, n} is the (fibered) torus knot of type (m, n). Abbreviate K{m, n}{p, q} to K{m, n; p, q}. (N.B.: there is no universally accepted standard notation for such iterated cable knots. In particular, instead of O{m, n; p, q}, Livingston & Melvin [2] write (q, p;n, m), and Miyazaki [3] writes (p, q;m, n).) Proposition. For any K, K{m, n} = K{m, sgnn}* O{m, n} is a 2m-gonal Murasugi sum, along a suitable Seifert surface F(m,sgn n) for K{m, sgnn} and a fiber surface D(m,n) for O{m, n}; for fibered K, F(m,sgn n) is a fiber surface. Proof. The case m = 1 is trivial. Take m > 1, sgnn = ±1. Let F ⊂ S be a Seifert surface for K. Construct a Seifert surface F(m,±1) from m parallel copies of F , each adjacent pair of copies joined, in order, by a 1-handle with a half-turn of sign ±. If F is a fiber surface, then so is F(m,±1) (see [6]); in any case ∂F(m,±1) = K{m,±1}. The proof is finished by contemplating an appropriate figure (see Figure 1). Proof of theorem. Livingston & Melvin [2] drew attention to the connected sum of iterated torus knots K := O{2, 3; 2, 13} ‖ =O{2, 15} ‖ =O{2,−3; 2,−15} ‖ =O{2,−13}. (Their motivation was a vague question [5] about relations among the concordance classes of the knots associated to complex plane curve singularities, and they gave an answer to one form of the question by observing that K is algebraically slice.) Miyazaki [3] showed that K is not ribbon. By the proposition, and the indifference of connected sums (of knots) to the location of the summation, K = (O{2, 3; 2, 1}*O{2, 13})‖ =O{2, 15} ‖ =(O{2,−3; 2,−1}*O{2,−15})‖ =O{2,−13} 1991 Mathematics Subject Classification. Primary 57M25.