Let \(\Sigma\) (K,1/N), \(n\in {\mathbb{Z}}\), denote the homology 3-sphere obtained by performing 1/n Dehn surgery on the knot K in \(S^ 3\). Various \(\Sigma\) (K,1/n) were known to have the property: (*) \(\Sigma\) (K,1/n) bounds a compact, orientable smooth 4-manifold which has a non- standard, positive definite intersection paring over \({\mathbb{Z}}.\)
The list of such \(\Sigma\) (K,1/n) is expanded upon here via clever arguments employing S. Donaldson’s work, constructions of cobordisms and the Kirby calculus. Results include: (1) if \(\Sigma\) (K,1/n) has property (*) then so must \(\Sigma (K',1/n)\) provided, under some planar projection, \(K'\) can be transformed into K by changing some positive crossings to negative crossings, (2) if \(\Sigma\) (K,1/(n-1)) has property (*) so must \(\Sigma\) (K,1/n), and (3) many \(\Sigma\) (K,1) are observed to have property (*) where K is an untwisted Whitehead double.
An immediate consequence of (2) is that if \(n<0\) then \(\Sigma\) (K,1/n) cannot have property (*) - since \(\Sigma (K,1/0)=S^ 3\). Those K occurring in (3) have infinite order in the smooth knot concordance group but are topologically slice.
The techniques employed also yield (4) if \(\Sigma\) (K,1) has property (*), then \(\Sigma\) (K,1/2n), \(n>0\), cannot bound a 1-connected smooth, compact, orientable 4-manifold having an even, definite intersection form.