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# Papers

These papers cover theory fundamentals as well as a variety of topics, including entanglement and dark matter.

## Fundamentals

This paper describes the core of the theory, and provides necessary background for the rest of the papers.

This paper reproduces the dynamics of quantum mechanics with a four-dimensional spacetime manifold that is branched and embedded in a six-dimensional Minkowski space. Elementary fermions are represented by knots in the manifold, and these knots have the properties of the familiar particles. We derive a continuous model that approximates the behavior of the manifold's discrete branches. The model produces dynamics on the manifold that corresponds to the gravitational, strong, and electroweak interactions.

C. Ellgen & G. Biehle

## Topics

Knot Physics describes the geometry of particles and fields. In a previous paper we described the topology and geometry of an electron. From the geometry of an electron we can construct a mathematical model relating its charge to its spin angular momentum. From experimental data, the spin angular momentum is $$ℏ/2$$. Therefore the mathematical model provides a comparison of electron charge to Planck's constant, which gives the fine structure constant $$\alpha$$;. We find that using only electromagnetic momentum to derive the fine structure constant predicts a value for $$\alpha^{-1}$$ that is about two orders of magnitude too small. However, the equations of Knot Physics imply that the electromagnetic field cusp must be compensated by a geometric field cusp. The geometric cusp is the source of a geometric field. The geometric field has momentum that is significantly larger than the momentum from the electromagnetic field. The angular momentum of the two fields together predicts a fine structure constant of $$\alpha^{-1} \approx 136.85$$. Compared to the actual value of $$\alpha^{-1} \approx 136.85$$, the error is $$0.13 \%$$. Including the effects of virtual particles may reduce the error further.

C. Ellgen

We describe entanglement and locality in Knot Physics. In Knot Physics, spacetime is a branched manifold. The quantum information of a system is encoded in the branches of the manifold. We show how that quantum information can persist despite the continual recombination of the branches of the manifold. We also note that the quantum collapse of state of the branches is non-local. That non-locality allows for non-local effects of entanglement without additional assumptions. We apply this description to the EPR paradox.

C. Ellgen

We describe features of cosmology in Knot Physics that have the characteristics of dark energy. Knot Physics assumes that spacetime is a branched 4-manifold embedded in a Minkowski 6-space. The cosmology of an embedded spacetime manifold is described by the expansion and contraction of the manifold in the embedding space. We show that the motion of the manifold in the embedding space contributes to the redshift of photons on the manifold. In this way, the embedded manifold model provides an alternative explanation for the redshift data that has been used as evidence for dark energy.

C. Ellgen

We describe dark matter in Knot Physics. Knot Physics assumes that spacetime is a branched 4-manifold embedded in a Minkowski 6-space. The theory has three metrics. The Minkowski space has the standard Minkowski metric $$\eta_{\mu\nu}$$. The second metric, $$\bar \eta_{\mu\nu}$$, is just the restriction of $$\eta_{\mu\nu}$$ to the spacetime manifold. In a previous work, we showed how mass and energy affect the curvature of $$\bar \eta_{\mu\nu}$$ , reproducing results of general relativity. The third metric, $$g_{\mu\nu}$$, is used to constrain the branches of the spacetime manifold. In this paper, we derive an approximate relationship between $$\bar \eta_{\mu\nu}$$ and $$g_{\mu\nu}$$. The relationship implies $$\bar \eta_{\mu\nu}$$ can have non-zero Ricci curvature without a massive source particle. We show how this result has many of the characteristics of dark matter.