WebThe property P conjecture is a corollary of the fact that for any non-trivial knot $K\subset S^3$, $\pi_1(S^3_1(K))$ admits an irreducible $SU(2)$-representation. In [33] , Kronheimer … WebThe second paper proves the so-called Thom conjecture and was one of the first deep applications of the then brand new Seiberg–Witten equations to four-dimensional topology. In the third paper in 2004, Mrowka and Kronheimer used their earlier development of Seiberg–Witten monopole Floer homology to prove the Property P conjecture for knots.
Definition: Theorem, Lemma, Proposition, Conjecture and …
Weband Mrowka in their celebrated proof of the Property P Conjecture [KM04]. More precisely, they proved that if K ⊂ S3 is a nontrivial knot, then there is a nontrivial homomorphism π 1(S3(K)) → SU(2), certifying that the surgered manifold is not a homotopy sphere. It is natural to conjecture WebRecall that the Property-P conjecture states that for any nontrivial knot K in S3, S3 K (m/n) has nontrivial fundamental group for every slope m/n = 1/0. The conjecture is an interesting special case of the Poincaré conjecture and remains a challenging open problem in knot theory and 3-manifold topology. See [15, Introduction] for a summary of mahavamsa written in which language
Property P -- from Wolfram MathWorld
WebAnother of Kronheimer and Mrowka"s results was a proof of the Property P conjecture for knots. They developed an instanton Floer invariant for knots which was used in their proof that Khovanov homology detects the unknot. Kronheimer attended the … WebNote that h 2 (p) =-p log p-(1-p) log (1-p) is the binary entropy function. Although there has been some progress in this line of work [ 2 , 3 ], this simple conjecture still remains open. There are also a number of variations of this conjecture. o2 arena play area