Computing the colouring invariant of a knot
The mathematical study of knots involves many different mathematical
techniques and has various facets. While in everyday usage the term
"knot" usually refers to a configuration of an open ended string, from
the mathematical point of view (the topological point of view to be
precise) these are not knots at all since they can easily be untied
and deformed into a straight piece of string. For mathematical
purposes, a knot is considered to be a continuous closed curve in 3-dimensional
space. Many of the keys questions concerning mathematical knots are
related to the problem of distinguishing two essentially different
knots.
Two knots are considered to be the same if they can be transformed
one into the other in 3 dimensional space without resorting to
scissors. For example, most people would agree that the following
pairs of knots are equivalent.
In each of the above diagrams the knots are represented by projections
onto the 2 dimensional plane (of the video monitor or paper if you print
this out). In 3 dimensional space the string never intersects itself
but in a projection this may appear to be the case. If the string is
represented as having a non-zero thickness then this will not usually
create any problem in interpreting the diagram. However, knots are
usually represented as lines without thickness (recall the mathematical
definition of a knot a curve in 3-space), If care is not taken to deal
with crossings, an ambiguous represention may result.
This is dealt with
by representing a crossing
so that the part of the string passing underneath the other part is
pictured as broken even though, in reality, it is not.
For example, the last two knots would be represented as follows
If a knot
represented in this way has k crossings then the string will
appear to be broken into k distinct arcs as well.
It is important to note that
the same knot can be projected onto the plane in various different
ways. However, by studying the provided examples
more closely, one quickly comes to the
conclusion that a knot is not essentially changed by performing
certain transformations to it.
For example, suppose that
this
is transformed to this
and then this
By repeating the mirror image of this sequence of moves one get the
following
from this
one gets this
and then this
This is known both of these are known as Reidemeister moves of type
I. One sees that the first pair of knots above can be
transformed into each other by a move of this type. Moreover, it
should be clear that moves of this type do not change the basic nature
of a knot. There are two other type of Reidemeister moves.
The second type is shown i nthe following sequence of moves.
A sequence of moves like this is known as Reidemeister move of type
II. A
Reidemeister move of type II will transform the two two knots of the second pair
above. There are also Reidemeister moves of type III. These are shown
below.
A Reidemeister move of type III will transform the last pair of knots
into each other.
Since it has already been noted that the
Reidemeister moves of type I, II or III will not essentially change
the nature of a knot, the same must be tru for any sequence of
Reidemeister moves. It is not a little surprising that these three
very simple type of moves are all that is necessary to exhibit that
two knots are the same.
Reidemeister's Theorem says that two knots are equivalent if and only if one can
be transformed into the other by applying a sequence of Reidemeister
moves. It is important to stress that Reidemeister's Theorem only
tells us that if two knots are equivalent then there is a sequence of
Reidemeister moves exhibiting this, but, it does not tell us what that
sequence is. In general, the problem of determining whether two knots
are equivalent is quite difficult.
One method for distinguishing between different knots is to assign
what are known as invariants to knots. As their name implies,
invariants in mathematics are invariant under certain
transformations. In the case of knots, invariants should be invariant
under the Reidemeister moves; in other words, an invariant of a knot
should not change its value when a Reidemeister move is applied to a
knot. If this is the case, the Reidemeister's Theorem immediately
implies that if two knots are equivalent then they have the same
invariant. Notice however, that there is no reaon to believe the
converse, knots with the same invariant need not be equivalent. An
easy way to see this is to assign to all knots the number 1. This is
clearly an invariant, but it is not very useful since it does not help
use distinguish any pair of knots.
The question of finding useful invariants though, is of considerable
interest.
One such invariant is known as the the colouring invariant of
a knot. To explain how this invariant is defined suppose that a
projection of a knot is given. It has already been observed that if
there are k crossings in this projection, then these also decompose
the string into k disjoint arcs. The integer m is
said to be a colouring invariant of this knot if it is possible to
colour the arcs using at least two different integers (in other words,
assign to each arc an
integer and so that at least two arcs are given different integers)
in such a way that at any crossing, if a and b
are the integers
assigned to the two arcs which end (or begin, depending on your point
of view) at the crossing and c is the colour of the arc which
passes on top at the crossing then a + b = 2c mod m. The
mod m is very important here because it allow knots to be
classified into several distinct equivalence classes. Without this,
there would be only two possibilities: either the knot can be
coloured or not. This would mean that knots are divided into only two
classes. Using colouring mod m allows a much finer analysis.
Finally, notice that the requirement that at least two colours are
used is also crucial since any knot can be coloured trivially using only one
colour so that all the equations are satisfied.
Of course simply calling something an invariant does not make it an
invariant. It is necessary to supply a proof
that the colouring invariant is an invariant before anything
further can be done with it. However, before looking at this proof, it
is probably instructive to look at an example of a knot and a
colouring of it.
In the projection shown, the knot at the left has 4 arcs and, of courses, 4 crossings as
well. The crossings have been labeled a ,
b , c and d while the
arcs have been coloured with integers from 0 to 2.
Each of the crossings corresponds to an equation which
must be satisfied by any colouring of the arcs. For example, at
crossing
a the arc passing below from the bottom is coloured
0, the arc passing below from the top is coloured 2 and the arc
passing on top of the crossing is coloured 1. At this crossing, the
equation 0 + 2 = 1×2 mod m must be satisfied.
But what is the value of m? It turns out that in this case
m = 3 will work. Since 2 = 1×2 mod m for any value
of m the equation
at crossing a is certainly satisfied. However, the equation
at crossing b uses arithmetic modulo 3 in an
essential way. the equation at this crossing is
0 + 1 = 2×2 mod m and it is satisfied mod m
only if m=3. The equation at crossing c is also
0 + 1 = 2×2 mod 3
while the equation at crossing d is
0 + 2 = 2×1 mod 3.
Use the colouring invariant to determine whether or not the following
two knots are equivalent.
When looking over the
provided solution pay attention to the fact that the solution is a
not perfect but this fcat is not hidden and is mentioned
in the summary.
Instructor
Juris Steprans
email address: steprans@mathstat.yorku.ca
Department of Mathematics and Statistics.
Ross 624 South, ext. 33952
York University
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