Knot Theory Presentation

This week at our weekly Queens’ Computer Science meeting, Eduard presented his Part II Project about Knot Theory. While Knot Theory is still a relatively new field of study and is largely being explored on a theoretical level, a significant application of Knot Theory has been to DNA in genetics.

The general problem of Knot Theory is to devise the best method to convert a knot, a twisted or otherwise manipulated circle in 3D space to a circle in 3D space, assuming that this can be done. Eduard’s project concentrates on trying out two different solutions to this problem and seeing which one more efficiently discovers the actions needed to result in a circle.

The first part of Eduard’s talk focused on the image processing aspect of his project. In order to transform a possibly unclear image to a graph in 2D, the pixels must be processed one at a time and converted to either black or white depending on whether or not the specific pixel represents a knot or the background. In addition to showing the image below, Eduard also performed a cool live demo in w he took a photo of a knot drawn on the white board with a marker and demonstrated the image processing, converting the imperfect hand-drawn image into a clear representation of the knot on the computer.

An example of an image containing a knot what needs to be processed before being used.

 

Once the redundant information in an image is removed, any bridges that were broken in the image need to be connected and the problem becomes how to figure out the center of each line to best be able to model the knot with a line and avoid various edge cases where the line may become stuck or curve back on itself while trying to trace the knot. The center of a knot segment can be computed by using circles of a finite radius, limited by the perimeter of the knot, sampled at various points in the knot and calculating the local maxima.

The knot can be represented in two different ways using Gauss codes, either by encoding the bridges with numbers and with + or – to represent either going over or under an intersection, or by using a three half-planes to represent the knot.

One method of using Gauss codes to represent a knot.

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A second way to visualize and represent a knot.

A finite number of the three Reidmeister moves can be applied to the knot diagrams to end up with a circle. Eduard plans to explore the two algorithms that solve the Knot Problem with various heuristics.

The three Reidemeister Moves which can be applied to a knot.

 

 

In addition to learning about the fascinating field of Knot Theory, we discussed some recommendations for giving good presentations, including committing the first thirty seconds of your talk to memory, since the audience will likely decide whether they will remain engaged within the first thirty seconds of a talk.

 

 

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