  The surgery unknotting number of a Legendrian link is defined as the minimal
number of particular oriented surgeries that are required to convert the link
into a Legendrian unknot. Lower bounds for the surgery unknotting number are
given in terms of classical invariants of the Legendrian link. The surgery
unknotting number is calculated for every Legendrian link that is topologically
a twist knot or a torus link and for every positive, Legendrian rational link.
In addition, the surgery unknotting number is calculated for every Legendrian
knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying
smooth knot has crossing number 7 or less. In all these calculations, as long
as the Legendrian link of $j$ components is not topologically a slice knot, its
surgery unknotting number is equal to the sum of $(j-1)$ and twice the smooth
4-ball genus of the underlying smooth link.
