Berge knot

From Wikipedia, the free encyclopedia

In the mathematical theory of knots, a Berge knot 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:

  1. K lies on a genus two Heegaard surface S
  2. 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.

References

  • 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, 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, 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 .
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