  Knotted vortices such as those produced in water by Kleckner and Irvine tend
to transform by reconnection to collections of unknotted and unlinked circles.
The reconnection number $R(K)$ of an oriented knot of link $K$ is the least
number of reconnections (oriented re-smoothings) needed to unknot/unlink $K$.
Putting this problem into the context of knot cobordism, we show, using
Rasmussen's Invariant that the reconnection number of a positive knot is equal
to twice the genus of its Seifert spanning surface. In particular an $(a,b)$
torus knot has $R = (a-1)(b-1).$ For an arbitrary unsplittable positive knot or
link $K$, $R(K) = c(K) - s(K) + 1$ where $c(K)$ is the number of crossings of
$K$ and $s(K)$ is the number of Seifert circles of $K.$ Examples of vortex
dynamics are illustrated in the paper.
