Conway sphere

In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot in the 3-sphere or 3-ball transversely in four points. It is essential if it is incompressible and boundary-incompressible in the knot complement.

References

Gordon, Cameron McA.; Luecke, John (2006). "Knots with unknotting number 1 and essential Conway spheres". Alg. Geom. Topol. 6: 2051–2116. doi:10.2140/agt.2006.6.2051. Retrieved 1 May 2014.
Lickorish, W. B. Raymond (1997), An introduction to knot theory, Graduate Texts in Mathematics, 175, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98254-0, MR 1472978

Knot theory (knots and links)
Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Whitehead (52 1) Borromean rings (63 2) L10a140
Satellite	Composite knots Granny Square Knot sum
Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (02 1) Hopf (22 1) Solomon's (42 1)
Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no. and problem
Notation and operations	Alexander–Briggs notation Conway notation Dowker notation Flype Mutation Reidemeister move Skein relation Tabulation
Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe Surgery theory
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