.H 1 "Elementary knot theory, a brief introduction"
.sp 2
.ft R
.po +6
.ll -6
.P
The theory of knots has had constantly waxing and waning popularity.  The 
popularity knots have enjoyed is most likely due to the fact that knot theory
really is the theory of knots: twisted and linked pieces of string.  
Also knots were a proving ground for a lot of the early work in topology.
The central question of knot theory is "when do two diagrams represent the
same knot?"  To answer this question we first must define some terms.
.P
A knot is always a piece of string with both ends attached (if the ends were
not attached there would be no theory, as any piece of string can be stretched
straight, but not all knots are equivalent to a simple loop).
The first point to be made is that all knots discussed here will be "tame 
knots".  A "tame knot" is a piece of string that has only a finite amount
of twisting.  Tameness is a property shared by all knots tied in actual string
(since all real string has non-zero thickness and finite length).  The
mathematical way to approach this is to study only knots that are built
by connecting a finite number of line segments (when altering a knot we
treat these as not being able to pass through
each other and having thickness) in 3-space (this is also
called a simplicial approximation).  Such a stiff
definition of a knot has the additional advantage that it is easy to draw
a diagram representing the knot.  The knot in 3-space is simply projected
onto a plane.  The resulting shadow is then a collection of line segments
(some possibly crossing).  Now since the knot is made of a finite number
of segments it is easy to see that there are only a finite number of points
on the projection where lines cross, it is also true that with a slight
change in the angle of the projection we can break a crossing that involves
3 or more line segments into several crossings involving only 2 line segments.
Furthermore, since everything is finite, it is alway possible to find a
projection such that all crossings involve only 2 line segments.  These
crossings can then be drawn such that we can see which segment passes under
which.  An example of the diagram of a simple knot, called the trefoil, can be
seen on the left.  It is customary to ignore the fact that knots are polygons
and draw the figures in the more relaxed fashion of the one on the right.
.sp
.DS
.PS
.so trefoil1a.pic
] with .nw at (0,0)
.so trefoil1b.pic
] with .w at K_trefoil1a.e + (0.5,0)
.PE
.DE
.P
As we said the central question is determining when two diagrams represent 
the same knot.  A concrete example would be to prove that one of the following
diagrams is equivalent to the trefoil pictured above and that one is not.
.sp
.DS
.PS
.so trefoil2a.pic
] with .nw at (0,0)
.so loop1.pic
] with .w at K_trefoil2a.e + (0.5,0)
.PE
.DE
The knot on the left can be deformed (without allowing pieces to pass through
each other) into the trefoil in three steps (illustrated below).  
Reidemeister proved that two diagrams represent the same knot if and only if
they could be deformed into one another using his 3 different types of
Reidemeister moves (and their inverses).  The moves are demonstrated as we
fix the trefoil.  First the string is pulled over a crossing 
(Reidemeister move number 3) then the string is pulled off another string
(Reidemeister move number 2) and finally the spurious loop is removed from 
the string (Reidemeister move number 1).
.sp
.DS
.PS
.so trefoil2b.pic
] with .nw at (0,0)
.so trefoil3.pic
] with .w at K_trefoil2b.e + (0.25,0)
.so trefoil4.pic
] with .w at K_trefoil3.e + (0.25,0)
.so trefoil1c.pic
] with .w at K_trefoil4.e + (0.25,0)
arrow from K_trefoil2b.e to K_trefoil3.w
arrow from K_trefoil3.e to K_trefoil4.w
arrow from K_trefoil4.e to K_trefoil1c.w
.PE
.DE
.P
Two diagrams that can be deformed
into each other obviously represent the same knot (since none of the
Reidemeister moves require a piece of string to pass through another
piece of string) but the usefulness of these moves is that Reidemeister
proved that two diagrams represent the same knot only if they can be deformed
into one another with the Reidemeister moves.  This theorem allows us
to study knots without using any topology.  In fact knot theory can be
reduced to a grammar problem in the following manner:   First label the n 
crossings in a given knot diagram with the labels 1 through n.  Then mark
an arbitrary (but consistent) directional arrow on all of the string 
and give each crossing a sign of "+" if the 
top string would be to point to the right if you were standing on the crossing
facing in the direction of the bottom string, else give the crossing a sign
of "-".  Signed crossings are demonstrated below:
.sp
.DS
.PS
.so arrowa.pic
] with .nw at (0,0)
.so arrowb.pic
] with .w at K_arrowa.e + (0.5,0)
.PE
.DE
.sp
Now walk along the knot one time and each time you encounter a 
crossing call out the sign, the label, and whether you are on the top or bottom
level.  Thus the following knot could be marked as shown and would yield
the sentence: "+1down to +2up to +3down to +4up to -5down to -6up to +4down
to +3up to +2down to +1up to -6down to -5up".
.sp
.DS
.PS
.so stevedore.pic
] with .nw at (2,0)
.PE
.DE
.sp
Then the Reidemeister moves could be rephrased as some kind of context
sensitive grammar.  Unfortunately, like many grammar problems, no algorithm
is known for generating a sequence of Reidemeister moves to transform one
knot into another.  And because the number of crossings do not help
determine an upper bound on the number of Reidemeister transformations required
brute force searching is not an effective method 
(it merely shows that the problem of determining if two diagrams are 
knot-isotopy problem is no harder than the
halting problem, which is not saying much).
Despite this the Reidemeister moves are very useful in knot theory.  The
most common use of them is to develop invariants.  Many papers present
algorithms that given a diagram calculate a polynomial from that diagram.
If the method of calculation is unaffected by all 3 Reidemeister moves then
it is easy to see that if two diagrams have different polynomials they do
indeed represent different knots (though the converse is often not true, 
nobody has yet found a simple invariant that proves diagram equivalence).
Many polynomials are able to differentiate the trivial loop from the trefoil.
