Simulating Quantum Loop Theory: Can S-Knots Be Represented Numerically?

In summary, there are many software and research papers on knot theory. It is an interdisciplinary field, with work done in the fields of materials science, biochemistry, and biophysics. There is much work being done to simulate knots numerically, and to detect them using software.
  • #1
kroni
80
10
Hello,
I am contacting you because I would like to know if there is a way to simulate quantum loop theory. Indeed, the S-Knots are much more complex objects than graphs because between the points there is a curve that can be knotted. S-Knots are graph embeddings in 3D and I do not see how such a theory could be simulated numerically. Graphs are easy to represented in computer science, but how encoded these knot. Since the theory is invariant to diffeomorphism, it is desirable to have a numerical representation that also take into account and retains only the knots of the graphs. How to set this type of structure? Is there no way to get rid of the knotting? I know the SpinFoam approach which can be simulated much more easily but it is not equivalent to the original theory.
I don't see any article speaking about, may be there is some big technological limitation ?

Thank you

Clement Deymier
 
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  • #2
Hi Clement,

Although your question is in the context of LQQ I think your main concern is around simulating knots.

A lot of work has been done on simulating knots. This work is wide ranging and interdisciplinary in nature. There is software to draw and detect knots and to carry out sequences of knot moves.

There is of course much mathematical work done on knot theory. However loads of work is done in the context of materials science - such as modelling polymers. At the moment there is work being done in biophysics and biochemistry in this area examining the folding and interactions of proteins and other materials such as dna and polysaccharides. The IEEE has also produced software and research on graphs and knots.

I hope this has given you some ideas for starting your own literature review in this area. Feel free to reply if you want more details or references.
 
  • #3
Interesting, what is LQQ exactly ?
Do you know a paper about knot simulation or the name of the software that detect knots ?
I think that type of algorithm must be very CPU time consuming due to complexity of algorithm on graph that are often n^2 or worst.
 
  • #4
Sorry for the typo - it should be LQG of course.

Knot software:
Good source of general knot theory software are:
and
For detecting knots see for example:
  • http://knots.mit.edu - KNOTS is a web server that detects knots in protein structures using the pdb or mmCIF format of the protein.
Whilst not about knot theory this work by Greg Egan inspired my own interest in on the numerical analysis of Quantum geometric operators and their spectra and I thought you might enjoy it :)
I hope all this gets you started at least.
 

1. What is quantum loop theory?

Quantum loop theory is a theoretical framework that attempts to unify quantum mechanics and general relativity by describing the fundamental building blocks of the universe as tiny loops or knots. It proposes that space and time are not continuous, but rather made up of discrete, quantized units.

2. How does quantum loop theory relate to S-knots?

S-knots are a type of knot that can be formed by the interaction of strings in quantum loop theory. These knots are thought to be the fundamental building blocks of the universe, and their properties and interactions are what give rise to the laws of physics.

3. Can S-knots be represented numerically?

Yes, S-knots can be represented numerically through computer simulations. These simulations involve using mathematical equations and algorithms to model the behavior and interactions of S-knots, allowing us to study their properties and make predictions about the universe.

4. What are the challenges in simulating quantum loop theory and S-knots numerically?

One of the main challenges in simulating quantum loop theory and S-knots numerically is the complexity and computational power required. These simulations involve solving highly complex mathematical equations and require a significant amount of computing power. Additionally, there are still many unknowns and uncertainties in quantum loop theory, making it difficult to accurately simulate certain scenarios.

5. How can the simulation of quantum loop theory and S-knots benefit our understanding of the universe?

By simulating quantum loop theory and S-knots, we can gain a deeper understanding of the fundamental laws and properties of the universe. This can lead to new insights and discoveries in physics, and potentially even help us develop a more complete theory of everything. Additionally, these simulations can also be used to test and validate theories and make predictions about the behavior of the universe at a microscopic level.

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