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

[kroni]

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

 

[David Horgan]

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.

 

[kroni]

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.

 

[David Horgan]

Sorry for the typo - it should be LQG of course.

Knot software:
Good source of general knot theory software are:

• http://katlas.org/wiki/Further_Knot_Theory_Software - this has links to all sorts of know theory software from Mathematica package knotTheory through to programs like knotplot https://knotplot.com/ and knotscape http://www.math.utk.edu/~morwen/knotscape.html,

and

• http://lewark.de/lukas/software.html - which has links to over a dozen programs.

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 :)

• http://www.gregegan.net/SCHILD/Spin/Spin.html

I hope all this gets you started at least.