Property P for the band-connect sum of two knots (Q1093210)

The author proves that the nontrivial band-sum of two knots has property P. This theorem implies immediately the affirmative solution to a long- standing conjecture (solved first by Scharlemann) that if a band-sum of two knots \(K_ 1\) and \(K_ 2\) is the unknot, then \(K_ 1\) and \(K_ 2\) are both the unknot and a band is also trivial. The proof of the theorem is short, but needs deep results due to D. Gabai.