Only finitely many alternating knots can yield a given manifold by   surgery
  We show that given a 3-manifold $Y$ there is only a finite number of alternating knots $K \subset S^3$ such that $Y$ can be obtained by surgery on $K$. A very similar but somewhat not complete statement has been obtained in a recent preprint of Lackenby and Purcell. 
Fyodor Gainullin
math.GT