The Structure and Enumeration of Link Projections
  We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition. 
Martin Bridgeman