Could the Universe be a Giant Knot?
Idea is that the state of the Universe is represented by a knot. The complex amplitude is given by a knot invariant. This is the sum of histories. A history is a series of slicings which return the knot to the unknot.
Knot states related by Reidmeister moves are equivalent.
The cosmological time of a knot is the 'unknotting' number.
One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1.
However, many of these may have very low amplitudes.
The amplitudes might be given by the knot polynomials, e.g,for some constant,k,q:
\[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]
Knot states related by Reidmeister moves are equivalent.
The cosmological time of a knot is the 'unknotting' number.
One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1.
However, many of these may have very low amplitudes.
The amplitudes might be given by the knot polynomials, e.g,for some constant,k,q:
\[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]
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