Low Dimensional Topology

July 4, 2014

Could a computer see a knot?

Filed under: Knot theory,Misc. — dmoskovich @ 5:36 am

I don’t know about you, but when I tell non-mathematicians what knot theory is, I often find myself telling a story about identifying a knotted protein by its knottedness- something about different proteins tending to be bendy to differing degrees, so that certain types of protein tend to form knots with higher writhe than others, and that this helps biologists and chemists to distinguish proteins which they would otherwise need a lot of time and money and an electron microscope to tell apart.

One major problem with this story, and with similar stories, is that the knot diagrams have to be photographed (and thus identified) by hand. The pictures are not always easy to interpret (e.g. distinguishing overcrossings from undercrossings):

A real picture of a knot.

Also resolution might be low, objects might be in the way…

This is a computer vision problem as opposed to a math problem- but wouldn’t it be nice if a computer could recognise a knot type from a suboptimal picture? If you could snap a picture of yourself standing in front of an 11n_{24} knot making bunny ears behind it, and your computer would automatically tag it with the correct knot type? Furthermore, wouldn’t it be nice if a computer could recognise your knot on the basis of many noisy pictures, perhaps taken from different angles?

In computer vision, there is a concept of a geon. A geon is a fundamental shape, such as a sphere or a cube, which a computer or the human brain can recognise from any angle even if the resolution is low and even if there are other objects in the way. The Recognition by components (RBC) theory asserts that vision is a bottom-up process which works by combining geons.

Geons have always been defined geometrically. A. Carmi suggested to me that topological geons should also exist. Indeed- a human can recognise a trefoil in any “reasonable” (i.e. fairly close to “minimal energy”) configuration, from any angle, even at low resolution and even if there are objects in the way. A computer ought to be able to do the same thing; and actually much more.

Computer vision is the most intensively researched field in applied computer science. It contains a huge body of research; all geometric and analytical as far as I know. Would it help to introduce some low-dimensional topology? Could topological geons such as knots and links help computers to see the world better? This would be a further manifestation of a “low-dimensional topology of information”!

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  1. Isn’t the major problem with the story that, as a matter of fact, very few proteins are knotted, and that biologists seeking to identify a protein have yet to use knot theory to do so?

    There was one effort going the other direction, using the known structures of known proteins to see which knots occur in nature. Alexander polynomials suffice to distinguish them, fwiw. http://knots.mit.edu/

    Comment by jdbatson — August 6, 2014 @ 1:37 am | Reply

    • The story has a number of problems, but I hadn’t known about this one (^_^).

      I was thinking mainly about the work of Peter Roegen, using finite-type invariants to distinguish proteins (conceptually this makes a lot of sense):
      e.g. http://www2.mat.dtu.dk/people/Peter.Roegen/poster2002.pdf

      Comment by dmoskovich — August 16, 2014 @ 7:45 pm | Reply

  2. Persistent homology is now being introduced to problems in computer vision. See e.g. Homological sensing for mobile robot localization

    Comment by ff — February 8, 2015 @ 3:03 pm | Reply

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