On certain non-trivial ribbon knots

Extract

Any ribbon knot K in S³ can be obtained from the unlink of r + 1 components {U₀, U₁,…, U_r}, for some positive integer r, by the simultaneous performance of r band operations (fusions), using disjoint bands. These bands may be assumed to run from U₀ to U₁, U₂, …, U_r respectively, and labelled b₁, b₂…, b_r accordingly. This paper investigates the question of what conditions on these bands guarantee that the knot K so constructed is non-trivial. Besides being of independent interest (see 1·2 (A) of (2), and the example below), this problem is related to the ‘slice implies ribbon’ conjecture (1·33 of (2)), to the triviality question for 2-spheres in the 4-sphere (section 10 of (4)), and other unknotting questions for surfaces in 4-space (see, for example, 4·30 of (2)). The main result proved here is that K is non-trivial if the union of the bands ‘geometrically links’ (in a sense made precise below) every U_i at most once, and at least one U₀ exactly once. The linking of the bands with U₀ does not affect the argument, so may be completely arbitrary.