Conway sphere

In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; such a curve does not always exist for an arbitrary knot diagram of a knot with a Conway sphere, but it is always possible to choose a diagram for the knot in which the sphere can be depicted in this way. A Conway sphere is essential if it is incompressible in the knot complement.[1] Sometimes, this condition is included in the definition of Conway spheres.[2]

A Conway sphere (black dotted midline) for the Borromean rings

References
  1. Gordon, Cameron McA.; Luecke, John (2006). "Knots with unknotting number 1 and essential Conway spheres". Algebraic & Geometric Topology. 6 (5): 2051–2116. arXiv:math/0601265. Bibcode:2006math......1265M. doi:10.2140/agt.2006.6.2051.
  2. Lickorish, W. B. Raymond (1997), An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98254-0, MR 1472978


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