In the mathematical theory of knots, a Berge knot (named after mathematician John Berge) or doubly primitive knot is any member of a particular family of knots in the 3-sphere. A Berge knot K is defined by the conditions:
in each handlebody bound by S, K meets some meridian disc exactly once.
John Berge constructed these knots as a way of creating knots with lens spacesurgeries and classified all the Berge knots. Cameron Gordon conjectured these were the only knots admitting lens space surgeries. This is now known as the Berge conjecture.
Progress on the conjecture has been slow. Recently Yi Ni proved that if a knot admits a lens space surgery, then it is fibered. Subsequently, Joshua Greene showed that the lens spaces which are realized by surgery on a knot in the 3-sphere are precisely the lens spaces arising from surgery along the Berge knots.
Yamada, Yuichi (2005), "Berge's knots in the fiber surfaces of genus one, lens space and framed links", Journal of Knot Theory and its Ramifications, 14 (2): 177–188, doi:10.1142/S0218216505003774, MR2128509.