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:
K lies on a genus two Heegaard surface S
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 space surgeries 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.
Berge conjecture
The Berge conjecture states that the only knots in the 3-sphere which admit lens space surgeries are Berge knots. The conjecture (and family of Berge knots) is named after John Berge.
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.
Further reading
Knots
Baker, Kenneth L. (2008), "Surgery descriptions and volumes of Berge knots. I. Large volume Berge knots", Journal of Knot Theory and its Ramifications, 17 (9): 1077–1097, arXiv:math/0509054, doi:10.1142/S0218216508006518, MR 2457837.
Baker, Kenneth L. (2008), "Surgery descriptions and volumes of Berge knots. II. Descriptions on the minimally twisted five chain link", Journal of Knot Theory and its Ramifications, 17 (9): 1099–1120, arXiv:math/0509055, doi:10.1142/S021821650800652X, MR 2457838.
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, MR 2128509.
Conjecture
Ni, Yi (2007), "Knot Floer homology detects fibred knots", Inventiones Mathematicae, 170 (3): 577–608, arXiv:math/0607156, Bibcode:2007InMat.170..577N, doi:10.1007/s00222-007-0075-9, MR 2357503.
Ni, Yi (2009), "Erratum: Knot Floer homology detects fibred knots", Inventiones Mathematicae,, 177 (1): 235–238, arXiv:0808.0940, Bibcode:2009InMat.177..235N, doi:10.1007/s00222-009-0174-x, MR 2507641.
Greene, Joshua Evan (2013), "The lens space realization problem", The Annals of Mathematics, 177 (2): 449–511, arXiv:1010.6257, doi:10.4007/annals.2013.177.2.3, MR 3010805.
External links
Two blog posts in the weblog "Low Dimensional Topology - Recent Progress and Open Problems" related to the Berge conjecture:
The Berge conjecture, by Jesse Johnson
Knot complements covering knot complements by Ken Baker
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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