Unlinking via Simultaneous Crossing Changes

Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers. Let L be an oriented link in s3with a generic projection onto the plane R~ . Let a be a short arc in R2 transverse to both strands of L at a crossing, so that the strands pass through a in opposite directions. Then the inverse image of a contains a disk punctured twice, with opposite orientation, by L . Define a crossing disk D for a link L in S3 to be a disk which intersects L in precisely two points, of opposite orientation. It is easy to see that any crossing disk arises in the manner described. Twisting the link q times as it passes through D is equivalent to doing l l q surgery on dD in S3 and adds 29 crossings to this projection of L . We say that this new link L ( q ) is obtained by adding q twists at D . Call dD a crossing circle for L . A crossing disk D , and its boundary do,are essential if dD bounds no disk disjoint from L . This is equivalent to the requirement that L cannot be isotoped off D in S3OD. Here we examine how a link can be turned into the unlink via simultaneous twists on disjoint crossing disks. In particular, we prove an analogue for pairs of crossing disks of the following theorem, a much more general version of which is proven in [ST, 1.41 (see also [Gal): 0.1. Theorem. If D is an essential crossing disk for the unlink L , then the link obtained by adding q # 0 twists to L at D is not the unlink. A moment's reflection suggests problems in finding an analogue for pairs of disks. Consider some examples: 0.2. Example. Suppose K1 and K2 are two essential crossing circles for any link L ,and suppose that K1 and K2 are unlinked in s3and bound an annulus in S3 L . Then adding q twists at K1 and -q twists at K2 leaves L unchanged. In particular, if L was the unlink before the twists were added, it will be the unlink afterwards. Received by the editors January 2 1, 1991. 1991 Mathematics Subject Classijcation. Primary 57M25; Secondary 57M40. The author was supported in part by a National Science Foundation grant. @ 1993 American Mathematical Society 0002-9947193 $1.00 + $.25 per page