>The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
arxiv.org/abs/0906.5177 Freedman, Michael; Gompf, Robert; Morrison, Scott; Walker, Kevin (2010), "Man and machine thinking about the smooth 4-dimensional Poincare conjecture", Quantum Topology 1 (2): 171?208, arXiv:0906.5177
で、コンピュータパワーで、結び目理論で計算したらしい 5.3 Results Computing the two-variable polynomial for K2 took approximately 4 weeks on a dual core AMD Opteron 285 with 32 gb of RAM. At this point, we haven’t been able to do the calculation for K3 . With the current version of the program, after about two weeks the program runs out of memory and aborts.
