Property P conjecture
Template:Inline-citationsIn mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P.
Research on Property P was started by R. H. Bing, who popularized the name and conjecture.
This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link. If a knot [math]\displaystyle{ K \subset \mathbb{S}^{3} }[/math] has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along [math]\displaystyle{ K }[/math].
A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.
Algebraic Formulation
Let [math]\displaystyle{ [l], [m] \in \pi_{1}(\mathbb{S}^{3} \setminus K) }[/math] denote elements corresponding to a preferred longitude and meridian of a tubular neighborhood of [math]\displaystyle{ K }[/math].
[math]\displaystyle{ K }[/math] has Property P if and only if its Knot group is never trivialised by adjoining a relation of the form [math]\displaystyle{ m = l^{a} }[/math] for some [math]\displaystyle{ 0 \ne a \in \mathbb{Z} }[/math].
See also
- Property R conjecture
References
- Eliashberg, Yakov (2004). "A few remarks about symplectic filling". Geometry & Topology 8: 277–293. doi:10.2140/gt.2004.8.277.
- Etnyre, John B. (2004). "On symplectic fillings". Algebraic & Geometric Topology 4: 73–80. doi:10.2140/agt.2004.4.73.
- Kronheimer, Peter; Mrowka, Tomasz (2004). "Witten's conjecture and Property P". Geometry & Topology 8: 295–310. doi:10.2140/gt.2004.8.295.
- Ozsvath, Peter; Szabó, Zoltán (2004). "Holomorphic disks and genus bounds". Geometry & Topology 8: 311–334. doi:10.2140/gt.2004.8.311.
- Rolfsen, Dale (1976), "Chapter 9.J", Knots and Links, Mathematics Lecture Series, 7, Berkeley, California: Publish or Perish, pp. 280-283, ISBN 0-914098-16-0, https://books.google.com/books?id=naYJBAAAQBAJ
- Adams, Colin. The Knot Book : An elementary introduction to the mathematical theory of knots. American Mathematical Society. pp. 262. ISBN 0-8218-3678-1.
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