(Reprinted from my old blog)
Some conjectures in mathematics are based on a large body of strong evidence. Other conjectures, it seems, are based on a lack of evidence to the contrary. The Berge conjecture appears to be of the latter sort. A simple closed curve in the boundary of a handlebody is called primitive if there is a properly embedded, essential disk whose boundary intersects the loop transversely in a single point. A simple closed curve in a Heegaard surface is double primitive if it is primitive in each of the handlebodies of the Heegaard splitting. A Berge knot is one that is isotopic to a double primitive loop in a genus two Heegaard splitting of the 3-sphere. The Berge conjecture states that if a knot in the 3-sphere has a non-trivial surgery that produces a lens space then the knot is a Berge knot.
The knots get their name from John Berge, who noted (in a never published preprint) that if one pushes a double primitive loop into one of the handlebodies, then there is a Dehn surgery on this isotoped loop after which the original loop bounds a disk in the new manifold. Because the original loop intersects a disk on the other side in a single point, the Heegaard surface in the new manifold can be destabilized, producing a Heegaard splitting of lower genus than the original. If one starts with a genus two Heegaard splitting, then the new manifold has a genus one Heegaard splitting. In other words, every Berge knot has a Dehn surgery producing a lens space. These are the only knots in the 3-sphere that are known to produce lens spaces and years of being unable to find any other examples has led some to conjecture that they are the only ones.
A fair amount of progress has been made on this conjecture, most of it coming out of work on Ozsvath and Szabo’s Heegaard Floer homology. The most recent stab from this direction comes from Baker, Grigsby and Hedden , who have announced a two part program to prove the conjecture, the first part of which has already been completed by Hedden . Here’s their idea:
Rather than looking at knots in the three sphere that produce lens spaces, they suggest looking at knots in lens spaces that have dehn surgeries producing the 3-sphere. Project the knot into a genus one Heegaard splitting of the lens space. This projection can be chosen so that the projection follows edges of a grid on the torus and so that horizonal edges are “above” the Heegaard splitting, while vertical edges are “below” (or vice versa). If one performs a lens space surgery on a Berge knot, the resulting knot in the lens space has what’s called grid number one: it has a projection consisting of a single vertical arc and a single horizontal arc. Conversely, if a Dehn surgery on a grid number one knot produces the 3-sphere then the image of this knot in the 3-sphere is Berge.
In the first paper, , the three describe a combinatorial way to calculate the Heegaard Floer homology of the knot based on a grid diagram. (This generalizes a method discovered by Sarkar and Wang  for grid diagrams of knots in the 3-sphere.) They use this to show that the rank of the Heegaard floer homology of a grid number one knot is equal to that of the lens space. (They call such a homology “simple”.) In the second paper, , Hedden shows that if Dehn surgery on a knot in a lens space produces the 3-sphere then the Heegaard floer homology of the knot is simple. This reduces the Berge conjecture to step two of their program: showing that every knot in a lens space with simple Heegaard Floer homology has grid number one.