On certain non-trivial ribbon knots

Abstract

Any ribbon knot K in S3 can be obtained from the unlink of r + 1 components {U0, U1,…, Ur}, 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, …, Ur respectively, and labelled b1, b2…, br 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 Ui 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.