Two salient efforts are prevalent in modern knot theory. Here we shall develop a convenient way of differentiating knots, as well as sequentially ordering and generating them. In both cases a modified form of Dowker notation is developed in the hope it generates an easily calculable and usable form.
A. Brief Reviews 0.5.) Functions (Maps)
A function which should be familiar to the reader is of the form y=a1 + a2x. The variables here are x and y and the coefficients a1 and a2 are considered unchanging or constant. The important notion is that for every “valid’ choice of variable x there is one (and only one) corresponding value, y.
The general definition of a function is thus any set of ordered pairs (x,y) such that if (x,y) is in the set as is (x,y1) then y = y1. (Thus no value x is associated with any more than one value, y.) [2, page 34] This general form of function may be thought of as having a domain of values which it “maps” to a range of values.
The function f(x) = ex has as domain any real number x and range any real number larger than zero as ex is larger than zero for every real number x. (This is due to the fact that e>0. [2, page 20 for more] Here ex is said to map x to ex.
When we refer to knots, we do not refer to knots tied in actual, physical string but rather to mathematical knots. Mathematical knots are as flexible and limber as the minds which ponder them. Generally speaking, a mathematical knot is a specific kind of function (with some important properties) mapping a two-dimensional circle into a three-dimensional space. In essence, any curve embedded in three-dimensional space which loops back upon itself and is closed in this sense is a knot. [1, page 1]
Notice the definition of knot would require us work with three-dimensional drawings which become unwieldy rapidly. As a consequence we work with two-dimensional projections of knots (the prior examples were all projections.) Thus represents one “strand” passing over another.
Definition If there a function exists which continuously maps one knot to another the two knots are equivalent. [1, page 12]
A consequence of this definition is that we may think of knots as infinitely stretchable or contractible (though we may not stretch them down to a single point.) [1, page 12] In essence, the following are all equivalent.
An interesting aside is the fact that all of the former were equivalent to the circle and are thus referred to as unknots. In fact, little work would be required to turn a few of them into ‘true’ knots . (We say ‘true’ to imply they are not equivalent to the circle.) While they are still knots they are called unknots as they are identical to the unchanged embedding of the two dimensional circle into space.
2.) Equivalence and Reidemeister Moves
The question of identifying two knots as distinct or equivalent seems therefore relevant as the prior definition of equivalence is unwieldy. What we would like is a way of easily deciding whether two arbitrary knots are equivalent or are fundamentally different. (What this ultimately means is that the statement that two given knots are different is equivalent to the statement that there does not exist a function which would continuously map the one knot to the other.) Fortunately, the local diagrams depicted in figure A.2.1 are known to be equivalent, and are called (suggestively) “Reidemeister moves.” (It is helpful to imagine the three-dimensional figures which these projections depict. With all of these moves we are simply moving strands over (or under) other strands which are not intersecting.)
Figure A.2.1 Reidemeister Moves [5, page 9]
Further, all of these projections “live” in space they may be ‘flipped’ 180 degrees to place strands over where they were under and vice versa (aka .) The prior process often generates figures which look different initially but are, in fact, equivalent. For this reason the following figures are equivalent. (This further equivalence is not associated with any Reidemeister move.)
3.) Dowker Notation
Before we may continue we must define knot orientation. If a knot has an orientation it is denoted with a (consistent) labeling of arrows along the knot’s projection and implies that while traversing along the knot we must follow in the designated direction and may not “cross back.” Such knots are said to be orientated and represent information about the mapping of the circle which is important to consider. [1, page 10] Further, any crossings having the following cross pattern are considered either positive or negative crossings, respectively.
So far we have discerned knots and defined them. A next logical step then would be to have some way of symbolizing knots in a written form rather than drawn. This is kin to writing “chair” to symbolize the object. One way of achieving this is use of Dowker notation. Dowker notation is an algorithm where by we traverse along an orientation of a knot labeling on the left side of each crossing the order we came to it. [1, Section 2.2] (For example, the first crossing we come across is labeled “1” on its left side, the next “2”, and so on.) We know we are done when every crossing has exactly two labels, as the knots we study all have 4-valent vertices and must eventually “loop back” to the place of origin. Following is an example Dowker notation of a trefoil.
By convention we write these down as a sequence of even numbers which it is assumed correspond to (1,3,5,…,n), as in 1 is assumed to be crossed with the first even number, 3 the second even number, and so on. [1, Section 2.2] Thus, for the former trefoil we have: