Thursday, October 20, 2011
Toward a Survey of Knot, Link, and Braid Algebraic Topology
Toward a Survey of Knot, Link, and Braid Algebraic Topology
L. Edgar Otto (The Pe Sla) October 20, 2011
I am not sure why someone wants to follow my blog in the sciences but I notice she follows some very interesting links such as:
Today I printed off the pages on knots and after reading a bit I had the desire to take a survey of all these alternative principles and philosophies of physics. This sort of classification probably is as many have said a doomed or limiting project- but we must somehow hold the wider totality of it in a nearly single grasp. I doubt with all the ideas of higher dimensions some still do not feel at home with that it all can in a sense be represented as a point. Now logically one could imagine and even prove this but would it go beyond our given world views on the matter.
In particular I see it works but it is not enough to imagine topological knots as if the transformations involved that these were as if immersed in a fluid.
From what I read the classification and description of knots if far from a settled issue. I find it interesting that Lord Kelvin first applied the idea which after all seemed so strange to me once- that matter is really knots in space- I mean that is hard to imagine if one can imagine space as a complete nothingness. So I would say in a simply connected world the topological dreams of Poincare would hold but it certainly is not enough for a general picture. I further see that embedding things as if higher knots in hyperbolic Minkowski spaces, even hyperbolic knots, is useful in the same sort of limited way- to think of four space in a six Minkowski space works but is that enough at the foundations? I have intuitively and informally said otherwise for these cases- yet I can admire things like Clfford's torus and the algebraic representation of Whitehead's links. Have fun you explorers before this sort of physics landscape seems a little dull to what is to come.
I doubt further that without a deeper understanding of how these things apply (such as the careful work of Kea) that knots alone and in this way will explain or resolve some of the issues in M theory.
Sure, one can extend the idea of a higher topology of Minkowski- much like I suspect is the intuitive foundations of Pitkanen's development. But this is not to be pushed away as some theory in the distance- for how else would the hint of relativity involving pair production even be imagined as an apology to explain how we can dismiss superluminal observations as if deep down they desire to either defend or demean the insights of Einstein.
So I, not only knowing how much I do not know and surprised as some of the things the world is not sure of where I myself had concerns in my questioning, have to at least explore what is on the net 101- If I only had more time, if my quiver in this quest game were not so close to empty of its arrows of time- I present this blog of thoughts that perhaps others can see about much lower level things as I have claimed no expertise or credentials- other than I strive to be the best human I can be in all this. I hope that for those of you with your interest you have some light hearted and happy moments along the way and are not discouraged by these temporary earthly and unjustified conflicts for what we great apes have achieved.
If I continue this survey- I will surely come back to post it. One can certainly be immersed in a sea of beauty without thinking about it, likewise our seas of sorrows.
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Knot Theory Survey (General Quasic Notation Principles): October 21, 2011
*1 if we do accept a sort of natural dimensional emerging state of the world of which it follows as per one omnic principle that atoms too can evolve- we imagine the point at some absolute beginning, exhausted infinitesimal of all such surreal spaces and numbers. Then the line or string of which we can eventually make knots, this being of the cardinality of aleph 1 (which in itself forms a continuum or a reference for other such quasi-transfinite continua).
*2 within this spacious view of natural dimensions, at any place in a particular knot in an abstract quasic motion not knowing of course if the path in a particular context is open or closed nor that the super-knot of aleph1 is minimally compact or generationally (quasically) more open in spaces so described between such knots- that inside such a place and moving we do not know (beyond a few structural steps) that we are in a line regardless of its absolute flat but intrinsic curvature or in a link or loop or knot or braid.
*3 in this sense we can say that for linear and limited super-knots (strings) these can form at any point in the dimensional evolving an invariant if independent structure- but the chirality of the knot in question is also at ground an invariant which fundamentally cannot be considered to have handedness- it follows that this property is only one of certain interconnections of dimensional motion functions where the distinctions are asserted or globally relative as if we impose an ordering with respect to time (but as I have not had the time or energy to work out the specific examples yet for this survey which amounts to a classification and an extension of the relations between various transfinite continua)for the physical as well the quasi-physical effects. Invariance to a great degree is to be seen then as topologicaly neutral when we use the quasic motion notation- furthermore, the ideas of crossings over can be distinguished from the properties of the quasic plane where these are needed- and they can be easily transformed into each other- and as a general idea of projective beyond the duality of points (lines) or circles entities that are thus neutral or seemingly at rest and in balance can in the various representations insofar as distance and proximity of points in the circuit of a knot-like object be in a dialectic choice of which is the primary space we call circular or linear. I suspect the quasic sequence of motion functions are unique and can only and intelligibly be distinguished when we apply the group concepts as they are intimately connected to the attainment of evolved dimensions where in the general flow of omnium the directionality and innate orientation of these structures so evolve the context too.
The description of such aleph1 knot systems that persist while dimensions evolve suggest that most of what our existing theories try to describe is this limited world of the first few layers of how such knots can evolve- these a sort of hidden ground by which we have missed the vast kinds of knots possible in higher space. We say for example that to cite say Hilbert in a work, the idea vaguely hinted at that we can use fractals when super-fractals are also needed, is to appear to be but a lecturing as if original thought rather than the intuitive idea of a fundamental breakthrough. In chess in the real game one is only as good as his opponent.
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