Double torus knot

Double torus knot

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]

The simplest example of a double torus knot that is not a torus knot is the figure-eight knot.

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?

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.

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