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Rob discussed his unique approach to knot theory, which he distinguishes from the traditional space of knots in R3 or E3. He explained that in his framework, knots can be made out of entities called E’s, which exist in a different space where E star E is not equal to E. This setup introduces a higher-level space, which he likened to a “place of places,” similar to type theory. Rob described how this allows for the concept of “concept sharing,” where crossings in knots are replaced with “sharings” of the underlying space. He demonstrated how labeling these sharings enables transformations of knots in this new space, connecting back to the original space. Rob concluded by suggesting that this different perspective on knots could be significant and popular.

Rob presented a follow-up video on knot theory, clarifying concepts from his previous video. He explained the distinction between higher and lower levels of an entity E, using a whiteboard to illustrate how these levels interact in a flexible space. Rob demonstrated how labeling in his system differs from the traditional Gauss code, allowing for more dynamic movement and flexibility in knot diagrams. He discussed the Reidemeister moves and how they relate to knot equivalence. Rob also mentioned that he had implemented related code on his website under the “Completion of Mathematics” section. The video aimed to provide further insight into his approach to knot theory.

Rob continued his series on knot theory by explaining the process of converting a knot diagram into Gauss code. He demonstrated how to label crossings with letters (A and B) and numbers (1, 2, etc.) to represent the direction of strand movement in three-dimensional space. Rob showed how to transform the labels from fixed positions to moving elements that can shift along the knot’s path without crossing in the original space, allowing for more flexibility in simplifying complex knots through reduction techniques. He concluded by explaining how this approach provides greater freedom in manipulating the knot diagram while maintaining the Gauss code representation.

https://youtu.be/BIB1qWnV20g

Rob created a video explanation of his mathematical concept-sharing theory, which introduces a new type of point called “E” that differs from regular points “P” by not collapsing into a single entity when combined. He demonstrated how E’s can exist at different levels (upper and lower) and can move relative to each other while maintaining their distinct identity. Rob also explained how numbers can concept-share using notation like “1,2” to represent partially shared numbers, and showed how this concept relates to a new number plane and Goldberg’s conjecture, which he has written extensively about on his website.

The explanation:

1. Crossings in R^3 or S^3 (positive space)
2. Label 1+,1-,ect. The locations of D-joinings/crossings (mixed space)
3. Place a, b at each D.
4. Move to negative space: move crossing now a sharing as we are capable of having 2 parts or more at a vertex with a and b always moving along.

1+,1-,ect. Are the specific locations in R^3 where we put the other concept sharing diagram back together.

But in the moving diagram they have some freedom. They can travel along as they are labels, reachable through moves of the diagram (that is moves are reversible) or they can stop at a specific pair of locations and the a, b pairs move on forward. This is still reversible as I can get back to this D, as I can reverse from forward motion. At the end, after I go all the way back I come back to the same location.

So 1+ and 1- label the crossing 1, where the original locations can be traced back from. I am placing the labels on specific locations. I mean that this is where the crossing is so that I can trace back to the original location which matches moves in this space then are fully reversible and take full advantage of the new freedoms. Then all these diagrams are equivalent and complete as long as we don’t cut the diagram or change the order of locations.

Then R1,R2,R3, isotopy have their equivalents as well. I have rotation of the locations, creation of labels, joinings, movements of joinings and labels through sharings. And that’s all (complete).

Let there be another diagram D(2) in R^3 and we wish to compare this to the original diagram. Move it to mixed space and then to negative space. Concept share it and one part of it moves off. If we can make a congruency between this and the other diagram in negative space then the two diagrams in positive space are also congruent.

So we need to look for a match of the labels, joinings when we simplify the diagrams.

If they are the same, then the information should be contained in one diagram. That is, movements of one diagram should be able to produce the second one. So we need to look at one diagram to see if we can produce another(using all moves available)

Let’s look at T. To cut down a lot of the moves, let’s apply a simplifying method.

Now I only need to apply S1,S2,S3 and isotopy(2) and I can obtain any diagram of T. Then reform the diagram to go back to R^3.

But the only new diagrams I can obtain will be through the application of the new moves to the T-invariant. That is all possible moves of the invariant. None yield the unknot or any achirality.

The new moves are rotation (which isn’t usually considered) and movement of labels and joinings through sharings. 

So here I am showing a way of seeing if the Trefoil is chiral. I take the 2 and 3 sharings out then see if I can bring them back after going through space in the different way with the 1 sharing. It turns out I can’t do that.

Here I am starting to use other diagrams of a knot. These come about as we are operating in negative space. That is the new space I created with negative distance. In that space I have the ability to pass through, also to move a series of connected parts ( the knot diagram itself) and create joinings (like (a3(b2)). These combined allow me to make new moves.

This is the oriented trefoil knot in three dimensional space.

So here I start with the Trefoil in three dimensional space. Then I move it to a mixed space which is described in the article on math concept sharing in this blog. I am sharing the concepts of a point and a number. We then can go back to positive space from here or on to negative space. The diagrams I draw after this are in negative space.