I finally got to read your paper with a fair amount of care to detail. I have not scored it yet. I read a hard copy last night and was not on line.
The following comes to mind with this. Given the four manifold M^4 a subregion D^2xT^2 is removed and the complement or dual of D^2xKxS^1 in S^3xS^1 is surgically inserted. It is common to think of spacetime as M^4 = M^3xR. So the manifold constructed from the knot K is
[math]
M_k =((M^3\setminus D^2\times S^1)\times S^1) \cup_{T^3} ((S^3\setminus(D^2\times K))\times S^1).
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On the left the R^1 in M^4 = M^3xR is replaced by S^1, and we can think of the S^1 as a periodic cycle with a real number line as a covering. Think of a wheel rolling on the real number line, or a spiral covering of a circle. In this setting the crux of the matter involves replacing a circle S^1 with a knot K. Physically this avoids topologies with circular time or closed timelike loops such as the Godel universe.
This substitution is then a type of cobordism. We think of there being a "tube" connecting a circle as a boundary at one end and the knot at the other end. This results in "crossings" or caustics of the tube, which suggest this image is viewed completely in higher dimensions. I attach an image of a situation where the knot is a trefoil. This is an interesting way to do cobordism. The boundaries of this space are a circle and the trefoil knot, and the relationship between these two is given by the Jones polynomial. The Jones' polynomial is a Skein relationship for a knot. The function W(C) = exp( i∫A•dx) is the Wilson line or loop integral for the valuation of a gauge connection. The expectation value is the path integral
[math]
\langle W(C)\rangle = \int D[g,A]W(C)e^{-iS}
[/math]
The element α = 1 - 2πi/kN, for N = mode number and k = momentum vector, and z = -2πi/k. Clearly then α^{-1} = 1/(1 - 2πi/kN). For k very large α^{-1} =~ 1 + 2πi/kN. The Skein relationship is then
[math]
\langle W(L^+)\rangle -\langle W(L^-)\rangle = 4\pi i/kN \Big(\langle W(L^+)\rangle + \langle W(L^-)\rangle\Big) = 2\pi i/k\langle W(L^0)\rangle.
[/math]
The trefoil is then under this polynomial equal to the circle plus two circles in a link, which is the Hopf link in the S^1 --- > S^3 ---- > S^2 series.
This enters into path integrals as
[math]
Z[g] = \int D[g] e^{iW(C)}e^{-iS[g]}.
[/math]
The cobordism then reflects a thin sandwich, to use Wheeler's terminology in Misner Thorne and Wheeler "Gravitation." The thin sandwich has a spatial surface of Cauchy initial data as the bottom slice of bread and Cauchy data on the top slice of bread or spatial surface. At the top slice the data corresponds to S^3\D^2xS^1, that is filled out into M^4\D^2xT^2, and the top slice is S^3\KxS^1. The action at the top and bottom of the thin sandwich is evaluated on the two topologies. This thin sandwich is in a sense "thick" if we think of the lapse function or diffeomorphism (homomorphism) connecting the two slices as R, and this is much larger than 2πr of the circles S^1 which "thickens" the spatial surfaces S^3\D^2xS^1 and S^3\KxS^1 into finely thin spacetimes. The action on the bottom is
[math]
S = \int_{M\setminus D^2\times T^2}dtd^3x \sqrt{-g}R + \int_{D^2\times T^2}dtd^3x \sqrt{-g}R,
[/math]
and at the top is is
[math]
S = \int_{(M\setminus T^2\times S^1)\times S_1)}dtd^3x \sqrt{-g}R + \int_{(T^2\times S^1)\times S_1)}dtd^3x sqrt{-g}R
[/math]
and the Willson loop is the ingredient that dictates the behavior of the
[math]
\delta S = \Big( \int_{(T^2\times S^1)\times S_1)}~~~ -~~~~ \int_{D^2\times T^2}\Big)dtd^3x \sqrt{-g}R.
[/math]
The knot topology or quantum group then dictates the quantum amplitude for the transition between the two configurations.
I don't know whether this connection to knots is strictly correspondent with the infinite number of "exotic" structures. It makes sense that one can have a fusion of knots in an arbitrary set of configurations. There is also the infinitely recursive knot-like topologies that Spivak discusses in his "Differential Geometry" books.
Cheers LCAttachment #1: knotcorb.PNG
