ON THE CONVEX HULL GENUS OF SPACE CURVES-t

LET K C R3 be a simple closed curve, and K be its convex hull. In [l], Almgren and Thurston define the (oriented) convex hull genus of K to be the minimal genus of an (oriented) surface contained in g and bounded by K. They give examples showing that even if K is unknotted both the orientable and non-orientable convex hull genus of K may be arbitrarily large. In 43 of this paper we show that their argument can be modified to apply to the class of almost-convex curves; these are roughly curves which are on the boundary of their convex hull except for small dips into the interior to avoid intersections. The formula we obtain gives a lower bound for the (oriented) convex huh genus of almost convex curves which is essentially independent of the topology of R3K; it is expressed in terms of the projection of K onto the boundary of K, and does not take into account which strand passes over the other. Moreover, we show in 42 that Seifert’s construction, appropriately modified, can be carried out within R, giving an upper bound for theconvex hull genus of K in terms of a spherical projection. For almost-convex curves, the upper and lower bounds coincide, giving an exact formula for both their orientable and non-orientable convex hull genus. The examples we obtain are much the same as those in [l] in the orientable case, but substantially simpler in the non-orientable case.