In this theory, we assume a few constraints on the spacetime manifold. One important result of those constraints is that the ways that the manifold can change topology are tightly constrained. If particles are knots on the manifold, then every production of particle/anti-particle pair is a change of topology of the manifold. Therefore a particular topology is only a viable particle topology if it can be created on the spacetime manifold subject to the constraining assumptions.
We prove that it is possible to produce pairs of knots of the form ℝ3#(S1×P2) subject to our constraining assumptions. The type ℝ3#(S1×P2) is relatively simple, analogous to a twist in the manifold. We then go on to show that those knots have the same properties as the elementary fermions. It remains to be shown that no other topology type is possible. It may be that there are other possible particle topologies and the only reason we see just the type ℝ3#(S1×P2) is that it has the lowest energy. While it is possible to mathematically eliminate many classes of particle topology, a complete list of possible particle topologies has not yet been determined.
The mathematical theory of knot theory has a very specific meaning for the word "knot". A knot is an embeddding of Sn in Sn+2. The constraint on the dimensions (an n-manifold in an n+2-manifold) is a tight constraint. For any other dimensionality, the embedding would not be knotted. For example, a knot of the form Sn in Sn+3 would spontaneously untie.
Every elementary fermion in this theory has topology ℝ3#(S1×P2). This is not a knot in the sense of knot theory. Also, this 3-dimensional topology can be embedded in ℝn for n≥5. A hadron, for example a proton, contains linked copies of ℝ3#(S1×P2), each of which is a quark. Those quarks will remain linked to each other only if the 3 spatial dimensions of the spacetime manifold are embedded in a 5-dimensional space. For that reason, we see that the dimension of the embedding space is perfectly constrained by the particle topology of hadrons.
In this theory, we need a word to describe particle topology. There are two reasons that we use the word "knot". The first reason is that the word is convenient and approximately describes the particle topology. The second reason is that, for hadrons, the 3-dimensional particle topology must be embedded in an 5-dimensional space, which is the same constraint (n=3 and n+2=5) that applies in the mathematical theory of knot theory.
In this theory, all elementary fermions are knots in the spacetime manifold that have topology ℝ3#(S1×P2). Leptons are elementary fermions and therefore they also have this topology. In this theory, we assume that the spacetime manifold is constrained by Ricci flatness, and we find that Ricci flatness strongly constrains the geometry of the particles. In particular, as charged leptons approach other particles, Ricci flatness forces the lepton radius to shrink. As the distance of approach goes to zero, the lepton radius also goes to zero. For this reason, leptons appear to be pointlike in collisions.
This description allows us to describe the spin angular momentum of leptons in familiar terms. For the case of pointlike leptons, spin angular momentum must be considered to be an inherent quantity that is not obtained in the normal sense of p×r, because the calculation would be meaningless if the lepton radius is r=0. In this description, radius is non-zero and the spin angular momentum can be calculated as a function of field strength from first principles.
The description of quantum mechanics here is very different from the usual description. We assume that the spacetime manifold is branched and we get all of our quantumness from that property. The viewpoint we adopt is that the path integral of quantum mechanics is useful because it is a convenient simplification of the complex dynamical system, which is the branched manifold. This is similar to saying that the heat equation is a convenient simplification of the transfer of heat energy through the vibrating molecules of an atomic lattice. The heat equation has certain pathological properties, for example energy is transferred faster than light speed. The path integral has certain pathological properties, for example various infinities that are difficult to tame. We hypothesize that the pathology results from modeling a discrete system as a continuous one. With the branched manifold, a single branch of the manifold is analogous to a single atom of the atomic lattice, and we do not try to know its random contribution. We just try to model the ensemble statistically. If we model the ensemble using a continuous approximation, the result reproduces the path integral of quantum mechanics.
So far all the key properties of physics (all four forces, quantum mechanics, and the known particles) have been shown to follow without requiring additional assumptions. There is a very short list of assumptions, even in comparison to the Standard Model, and the consequences of those assumptions have been adequate to describe every physical phenomenon that has been attempted. Interestingly, over the course of the development of the theory, the set of assumptions has become ever smaller as previously distinct descriptions were shown to be unified by ever simpler models.
Because this theory is relatively new, we find that most people prefer to discuss it with their friends and colleagues first. The videos and presentations are designed for sharing among people who are new to the theory. We have had quite a lot of time to think about the theory and have been able to address most of the questions that people have asked. If you have any questions or you are curious about future research possibilities, please feel free to email.