A note on unknotting number

Abstract

For a classical knot K in the 3-shere, the unknotting number u(K) is defined to be the smallest number of crossing changes required to obtain the unknot, the minimum taken over all the regular projections. This dependence on projection makes the unknotting number a difficult knot invariant. While some algebriac methods exist to give a lower bound for u(K) the unknotting number for approximately one-sixth of the 84 knots with nine or fewer crossings remains unddetermined, see [9] or [7]. For an upper bound, one is usually forced to intellgently experiment knotting various projections of the knot under study. Usual practice is to work with a minimal crossingprojection; indeed, it has long been a ‘folk’ conjecture that the unknotting number is realized in such a projection ([5], p. 21). This note shows by example that this conjecture is false. This remarkable knot is rational, i.e. a 2-bridge knot, and hence alternating. Thus there is also the surprising result that the unknotting number of an alternating kniot is not necessarily realized in a minimal alternating projecting.