According to legend, Alexander the Great faced a challenge, where his success prophesied his eventual triumph over lands in Asia. Alexander considered untying the knot, but chose instead to use his sword to cut the knot, revealing its ends. After this, the knot could be easily unwound. The question here is, if the Gordian knot cannot be untied except by revealing/creating ends, can it be a knot?
Knots are created by a linear combination of moves done upon strings that result in a series of bends. Some “knots” are trick knots – a slip “knot,” without much tension on either end, quickly unravels. Here we will define these trick knots as not true knots. A knot must hold weight for extended periods, resisting capsizing or slipping (both of which result in coming untied). An example of knot-steps that would not count would be two strings wound around each other. If either string is pulled upon, unless there is an exorbitant friction coefficient, the two strings will come apart, back to their original condition.
What our project looks at is how to go from a completed knot back to the individual string(s) that it originated from. We hope to develop a method that covers knots in the general case. Our theory frames knots such that they only result from linearly independent ends of the strings (working end and standing end), with linear moves in which order matters (though this last part we will investigate more theoretically than factually), and only two strings. Therefore, the solution to the matrix that results in the knot in the first place will be the inverse matrix formed by each step of tying the knot. (Consider each transformation on the string[s] as a standard matrix.)
The reverse to this theory is our prediction that the only way a knot cannot be untie-able is if the string’s ends are not linearly dependent – visually where a string meets itself again, it is fused at some level. Furthermore, we might even suggest that a knot that does not appear untie-able would be the monkey’s fist, where one could nearly hide the ends within the knot itself. Therefore, the Gordian knot fails to be a knot in its inability to be untied.
The expansion of this theory leads to chains of operations. When you repeat certain processes, you get knots on top of knots. Because knots have a standing end and a working end, when this process is repeated, you get a number of stacked knots. This number can be considered an eigenknot (or scalar of a knot). If you think about a plane in 2D space, you would get the same knot repeated at a larger y. The thickness (3D component) here is not necessary. Continuing the process infinitely to tie a square knot will result in a series of square knots until either the working or standing ends run out.
In conclusion, we hope to show that knots can be represented in a matrix form of unique operations that will row reduce to result back into a solution with a positive number of possible answers. What will be interesting to see is if there are more than unique answers – we certainly expect this from larger knots comprised of prime knots, but we also consider that there is more than one way to untie, and furthermore to tie, a square knot (and that furthermore the ease of untying when pulling down on one loop is what led to the rise of said square knots).
Our main problem will be finding a way to model knot interactions (steps) in a way that gives sensical answers for this class. Certainly, we can think about each of these interactions as symmetric to things we do in linear algebra, so it will be necessary to combine our two sciences to create something that is inherently linear algebraic rather than merely representative of what we have been doing.