Today we explore the topic of Knot Theory. Most of us have a conventional idea of what a “knot” is; and those who were once Boy Scouts may have a more formal idea. But in mathematics, a knot has a very formal defintion: an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). An example of a mathematical knot, the “figure eight knot”, is illustrated below:
Basically, it is a continuous curve in three-dimensional space that loops back on itself, crossing an arbitrary number of times but never cut or spliced. It relates to the conventional notion of knot as a piece of string connected at the ends.
Knots relate to other topics that have been explored here at CatSynth, such as Lissajous curves.
Many knots, when projected onto a two-dimensional plane form Lissajous curves.
It also relates to our interest here at CatSynth in highway interchanges, such as this the intersection of I-105 and I-110 in Los Angeles:
Knots (and tangles) can be arbitrarily complex with twists and crossings. But there is order in this twisty world, and indeed knots have properties analogous to numbers, such as equivalency and prime decomposition.
Two knots are considered equivalent if one can be converted to another by simple scaling (stretching or rotating), or performing one of several Reidemeister moves, twisting or untwisting in either direction, moving one loop (or segment) completely over another, or move a string completely over or under another crossing. Basically, this is any operation you can do on a closed string without cutting or splicing it.
Here at CatSynth, we are quite familiar with Reidemeister moves, as they seem to occur spontaneously on our audio cables:
Even more interesting is how knots can be decomposed into prime knots. Just as any integer can be expressed as the product of prime numbers, any knot can be expressed as the “sum” of prime knots.
Here is a chart of the first 15 prime knots:
Here, the prime knots are grouped by the number of crossings. For example, the trifoil knot (second from the left on the top) has three crossings. The circle is a degenerate case, known as the “unknot”, with zero crossings. As the number of crossings increases, the number of possible prime knots also increases. For example, there are seven unique prime knots with seven crossings.
For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table.
|number of prime knots
This sequence (formally listed as A002863), appears to grow exponentially. Indeed, results by Welsh show a lower bound of 2.68 for the exponential base, and an upper bound of 10.40 due to Stoimenow. However, as far as I can tell, there is know known analytical formula for this sequence, and the values for n=17 and above are not known.
I find such sequences of numbers fascinating. Where to they come from, and how does one figure out the next value? In the case of prime knots, these appear to be open problems.
For more information, Giovanni de Santi has an excellent introduction to the theory of knots. Another paper by [url=http://algo.inria.fr/bsolve/constant/constant.html]Steven R. Finch is a resource for advanced analysis of knots and tangles, including more on counting prime knots.