In the relatively new mathematics known as "Knot Theory," there are a few important questions that must be asked right from the beginning, upon being faced with analyzing a given knot (generally portrayed as an image of a rope or string looping in and out of itself, following a complex sequence of over/unders):
Does a Knot Even Exist?
Is there an actual knot being shown here? That is, could it be possible that what appears to be a knot is nothing more than a series of loops which, when stretched out, do not form a knot at all?
There are a few methods to determining this fact - some of which are deceptively simple - most of which have to do with determing the sequence of over/under passes.
Tricoloration is one of the most fundamental and simple methods of determing a knot. In essence, to tricolor a knot requires applying a different color to each section of knot, beginning at any point, coloring along the knot with one color until it passes under another section. At that point, a second color is begun, following until another section is passed under. A third color thus begins until a third line is passed under, then the process begins over with the first color.
At the end of this process, the "tricolorability" of a knot can be verified if at every passing either 1) all three colors are present or 2) all sections are of the same color.
The Reidemeister Moves
After the determination is made regarding a knot's existence, the knot may be simplified. To do this, knot theorists familiarize themselves with all the ways that a knot might be deformed without changing its essential form (changes such as these are said to be "istopically invariant").
The possible "moves" that can be made to a knot are known as the "Reidmeister Moves," after their discoverer, Kurt Reidmeister (1893-1971). These moves include twisting the rope, pass an entire loop over another, or pass a strin completely over or under another crossing.
When a Reidmeister move is performed, it surely complicates the appearance of the knot, but it doesn't effect the actual knot from a topological perspective (in other words, the degree of the knot remains the same) Reidmeister moves can be "undone" and the knot will remain the same, though will appear simpler.
Classify the Knot
Knots are classified, generally, based on the minimum number of their crossings (that is, the number of crossings after discounting the Reidmeister moves).
The simplest knot, known as a "trefoil" knot (or, in common parlance, an "underhand" or "granny" knot) has a crossing number of three. Anything less than that would be classified as an "unknot."
Knots can further be defined by whether or not they are "prime" knots (that is, whether their crossing number is a prime or not), and whether or not they can be "transformed" into other knots without changing the crossing number.
Knot theorists have also classified knots by "handedness," that is, they have differentiated between identical, yet opposite knots, and much work has been done on determining if left handed knots might be able to "cancel" right handed knots.
Discover Mathematical Rules
Much of knot theory comprises the exceedingly difficult search for algoriths regarding transformations and determinations of knot classifications. Mathematicians search for more and more intricate and general algorithmic functions that help to solve knot problems (such as the Jones Polynomial and the Alexander Polynomial).
Of course all of this is only scratching the surface and providing just some of the most basic elements of knot theory in a specific sense. There remains much more to be said, especially as knot theory continues to grow into a more vital and important mathematical theory.
See Also:
References:
"Knot." Wolfram Mathworld.
Gardner, Martin. "The Colossal Book of Mathematics." W.G. Norton. 2001.
Isaac M. McPhee - Isaac McPhee was born as a human child in Mt. Vernon, WA, c. 1982; he currently resides in the bustling heart of New York City where he ...
