Double torus knot

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A double torus knot is a closed curve drawn on the surface called a double torus (think of the surface of two doughnuts stuck together). More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus two handlebody in S³. If a link is a subset of a genus two handlebody, it is a double torus link.[1]

While torus knots and links are well understood and completely classified, there are many open questions about double torus knots.

Two different notations exist for describing double torus knots. The T/I notation is given in F. Norwood, Curves on Surfaces[2] and a different notation is given in P. Hill, On double-torus knots (I).[3] The big problem, solved in the case of the torus, still open in the case of the double torus, is: when do two different notations describe the same knot?

[edit] References

  1. ^ Dale Rolfsen, Knots and Links, Publish or Perish, Inc., 1976, ISBN 0-914098-16-0
  2. ^ Topology and its Applications 33 (1989) 241-246.
  3. ^ Journal of Knot Theory and its Ramifications, 1999.