A Knot is a simple closed curve in threedimensional space, where
``simple'' means the curve does not intersect itself and ``closed'' means
it ends where it starts, i.e. it forms a closed loop without loose ends. We may
think of a knot in this sense as the result of taking a curve segment, tying
a knot in it, and then joining the ends together so the knot cannot come
undone. We usually describe knots by drawing pictures called knot diagrams.
A knot diagram is a projection or shadow of a knot on the plane, with
the restriction that each point on the diagram is the shadow of no more than
two points on the knot. Each double point, i.e. each point which is the
shadow of two points on the knot, is called a crossing, and we draw
crossings on the knot diagram with breaks to show which strand crosses
over and which strand crosses under.
A choice of a direction to travel around a knot is called an
orientation
of the knot, and a knot together with a specific choice of orientation is
called (not surprisingly) an oriented knot.
When we consider oriented
knots, we find that there are two types of crossings, corresponding to
righthanded
twists and lefthanded twists. We will call one type of crossing
positive (a righthanded twist) and the other type
negative (a lefthanded twist), and the signs + and  will be
attached to positive and negative crossings respectively.
As any scout or sailor knows, there are many different kinds of knots.
In knot theory, we would like to be able to tell when two knots are
the same and when two knots are different. In particular, we would
like to know what are all of the possible types of knots.
Moving a knot around in space does not change what kind of knot it is  in
order to get a different knot, we have to cut the strand, tie a different
knot in it, and rejoin the ends.
In general, it is difficult to tell just by looking at two knot
diagrams whether they represent different knots or the same knot.
The intuitive idea of ``moving a knot around in space'' corresponds to
the mathematical idea of ambient isotopy; we are free to stretch, bend,
and otherwise deform the knot and as long as we don't tear it or intersect
it with itself, we don't change what type of knot it is. When one knot can
be obtained from the other by an ambient
isotopy, the two knots are called ambient isotopic. In other words,
to say that two knots are the same type is simply to say that they are
ambient isotopic.
In terms of knot diagrams, ambient isotopies translate to sequences of
plane isotopies and Reidemeister moves. A plane isotopy allows us
to move a curve around in the plane, stretching or bending it however we
want so long as we don't intersect it with another curve or move it out
of the plane. If we want to move a piece of the knot from one position
in the plane to another, there are four kinds of situations that can
occur, based on the kinds of things can be in the way:
 nothing (plane isotopy)
 the arc we want to move (move I)
 a different arc (move II)
 a crossing (move III)
In the first situation, where nothing is in the way, we simply do a plane
isotopy to move the arc to where we want it. In the other three situations,
we apply the following Reidemeister moves:
In these pictures, we mean that if we have two knot diagrams which are
identical outside of the pictured area and the only change from one
to the other occurs as pictured, then the two diagrams represent the same
knot.
A sequence of plane isotopies and Reidemeister moves will be called an
isotopy, and two knot diagrams connected by an isotopy will be
called isotopic.
There is a theorem which says that two knot diagrams arising from ambient
isotopic knots are related by a series of plane isotopies and Reidemeister
moves, and conversely plane isotopies and Reidemeister moves can be "lifted"
to obtain ambient isotopies in knots. So, if we can produce a series of
these moves that change one knot diagram into another, then we will have
shown that the two diagrams represent the same knot. Not being able to
produce such a sequence, however, doesn't tell us anything  we may not
have been clever enough to find the right sequence, or the sequence of moves
(if one exists) may be very complicated, involving thousands or even millions
of moves.
So, to prove that two knots are not the same, we have to be more clever.
The usual technique that knot theorists use is to devise knot
invariants, which are numerical or algebraic quantities associated to
particular knot diagrams in such a way that doing Reidemeister moves on the
diagram does not change the numerical quantity. If a numerical quantity
calculated from a knot diagram stays the same under Reidemeister moves
(and plane isotopies), then any two diagrams of
the same knot will give us the same value for that invariant; in particular,
if two knot diagrams give us different values for some particular invariant,
then they can't represent the same knot. This gives us a way to prove that the
two diagrams in question represent different knots.
However, if two knot diagrams give the same value for an invariant, it
does not generally mean that the two knot diagrams represent the same
knot; we have to either check other invariants to see if another invariant
might distinguish the knot diagrams or find a sequence of isotopies before
we can determine whether the two diagrams represent the same knot.
Looking for sequences of isotopies and calculating invariants can both
be timeconsuming tasks and, as in all complex calculations, doing them
by hand can introduce human errors quite easily. Thus it makes sense to
look for a way to get computers to do these jobs for us.
There is an obvious problem, however: the way we usually describe knots
is by drawing pictures, our knot diagrams mentioned above, and while these
are humanfriendly in that they give us a clear visual impression of the
knot in question, knot diagrams are not very computerfriendly. Thus
if we are going to use computers to do calculations on knots, we need to
find a more computerfriendly way of describing knots.
One way to do this is to use Gauss codes. We create a Gauss code
from a
knot diagram by first labelling each crossing with a number. (Let's assume
there is only a finite number of crossings; knots with infinitely many
crossings are called wild knots and can be problematic, so we'll
restrict our attention to tame knots, i.e. knots with only a finite
number of crossings). We then choose a starting point anywhere we like on
the knot and follow the chosen direction around the knot and write down
the order in which we encounter each crossing along with whether we cross over
it or under it and what its local orientation is (+ or ), keeping on until
we arrive back at our starting point. Thus a Gauss code is a sequence of
crossing labels listed with an ``O'' or a ``U'' and a ``+'' or a ``''.
Since we encounter each crossing exactly twice, once going over and once
going under, each label will occur exactly twice in our Gauss code, once
with an ``O'' and once with a ``U''. The sign of a given crossing is the
same for both occurrences of its label in the Gauss code.
Example: This is the trefoil knot (also commonly called the
overhand knot),
which most of us use in tying our shoes. It has the Gauss code
O1U2O3U1O2U3, as you can see by starting at the indicated point and
following the knot along, writing down the crossings in the order you
encounter them and whether you cross over or under and whether the crossing
is positive or negative, until you return to the starting point. Note that
all three crossings here are negative.
Great! Now we have a way of specifying knot diagrams that uses only ordinary
keyboard characters and can fit on as little as a single line. In order for
Gauss codes to be useful to us a way of describing knots, we have to check
that we can recover the ordinary knot diagram from the Gauss code. Of course,
the exact code we get depends on our choice of starting point, but it is
easy to see that choosing a different starting point will simply move some
labels from the end to the beginning of the code.
A much more serious concern arises when we consider taking a arbitrary
Gauss code and trying to construct a knot diagram from it: as it turns out,
there are some Gauss codes that don't give us knot diagrams at all! Try to
draw a knot diagram with the Gauss code O1+U2+U1+O2+, for example; you'll
find that you have to introduce an extra crossing to join the end back up
to the starting point.
This is bad because in a Gauss code we are supposed to have already listed
all of the crossings; we can't add additional crossings without
changing the Gauss code.
Virtual Knots
Gauss codes are more computerfriendly than knot diagrams for specifying
knots, but some Gauss codes do not correspond to ordinary knots. Nonetheless,
Gauss codes are potentially very useful in doing calculations,
so we don't want to abandon them; as it turns out, we don't have to. The Gauss
code O1+U2+U1+O2+ may not represent a real knot, i.e. a simple closed
curve in threedimensional space, but it does represent something called a
``virtual knot.''
This situation is very much like when we first consider the question
``what is
the square root of negative one?'' Our initial idea of what a ``number'' is
based on the idea of cardinality, like the number of bananas in a bunch; but
when we start thinking of numbers as solutions to algebraic equations rather
than as cardinalities, we find that there are some algebraic equations which
none of the numbers we're familiar with satisfy, like x^2 = 1
(or even x+1=0).
In order to solve these equations, we define a new concept of ``number''
that extends our old idea, so our previous idea of what a ``number'' is is
a special case of our new, more general idea. We can do this as long as
our new definition is consistent with our old one. In the number case,
defining a new number i by the equation i^2 = 1 allows
us to solve all of our algebraic equations. We would like to extend our idea
of ``knots'' in a similar way so that every Gauss code represents a ``knot''
of some type.
The way we do this is to start by considering the Reidemeister moves in
terms of Gauss codes. More precisely, we'll write our Gauss codes in a
slightly more humanfriendly (i.e. geometric) form, using what we call
Gauss diagrams. We take a circle and write our Gauss code
counterclockwise around the circle, connecting the two occurrences of
each label with an arrow pointing toward the undercrossing (the one with
the ``U'') and decorating each arrow with a ``+'' or ``'' for local
orientation.
In terms of Gauss diagrams, the Reidemeister moves now look
like:
Once again, these pictures mean that the two Gauss diagrams are identical
outside the areas pictured. The epsilon stands for either sign +1 or 1
(or just + or ). So,
for example, move I means that we can introduce an arrow with either sign
anywhere on the diagram provided that no arrowhead or tail lies between the
head and tail of the one we are introducing, and also that if there is an
arrow with no arrowheads or tails between its head and tail, we can
delete it.
We now define virtual knots to be equivalence classes of
Gauss diagrams (or Gauss codes, whichever form you prefer) under the
three above moves. In other words, two Gauss
diagrams represent the same virtual knot if there is a sequence of the
three above moves that changes one diagram into the other. Such a
sequence will be called a virtual isotopy. A Gauss diagram which
represents an ordinary knot diagram is called classical or
realizable, and a Gauss diagram which does not represent a
real knot (i.e. one for which we have
to introduce additional crossings to make the diagram work) is called
nonclassical or nonrealizable.
This new definition is consistent with our old idea of what a ``knot'' is,
since any sequence of ordinary Reidemeister moves on knot diagrams translates
to a sequence of the above moves on Gauss diagrams. However, we can turn a
realizable knot into a nonrealizable one by using the above moves: an
unknotted diagram (one with no crossings) can be changed into the
nonrealizable diagram U1+U2O1+O2 by a type II move, for example.
We might then be worried that two classically inequivalent classical knots
may be equivalent as virtual knots, but this is not so. In fact we have:
Theorem [2]: Two virtually equivalent classical
knots are classically equivalent. In other words, if two realizable Gauss
diagrams are virtually isotopic, then they have realizations (i.e. they
represent real knot diagrams) which are connected by a sequence of ordinary
isotopies.
So, we have extended our notion of what a ``knot'' is to include virtual
knots, abstract Gauss codes which do not correspond to ordinary knot diagrams.
Gauss codes and diagrams are fine for computers, but we humans prefer
ordinary knot diagrams since they show us clearly what the knots look like,
so we would like to be able to draw knot diagrams for virtual knots. We can
do so if we include a new type of crossing, called a virtual crossing.
Virtual crossings are virtual because they aren't really there; they are the
result of drawing nonplanar diagrams in the plane. Virtual crossings do not
have a sense of over or under, and they do not represent selfintersections
either. Virtual crossings don't
show up in Gauss codes or diagrams; they're just a trick we use to let us draw
``knot diagrams'' for virtual knots. So to draw a virtual knot diagram
from a Gauss code, we draw the crossings and connect the strands in the order
specified by the diagram, introducing virtual crossings whenever we have
to.
A virtual crossing looks like this:
Any two virtual knot diagrams (knot diagrams including real and possibly
virtual crossings) with the same real crossings (including orientations)
occurring in the same order will have the same Gauss code, so they represent
the same virtual knot. A Gauss code or diagram tells us how many real
crossings (also called classical crossings) there are in our virtual knot,
which order we encounter them in, and what their orientations are. In other
words, a Gauss code determines how a small neighborhood of each real crossing
looks and in which order the segments in those neighborhoods get connected.
Apart from that, we are free to draw the arcs outside of those neighborhoods
however we want, using virtual crossings whenever one arc encounters another
(or itself). In particular, to change one virtual knot diagram into another
equivalent virtual knot diagram, we need to be able to move an arc which
contains only virtual crossings arbitrarily around in the diagram.
To accomplish this, we need analogues of the Reidemeister moves for virtual
crossings. As before, let's consider what kinds of things can get in the
way of moving an arc with only virtual crossings to another location:
 nothing (plane isotopy again)
 the arc itself (move VI)
 another arc (move VII)
 a virtual crossing (move VIII)
 a real crossing (move V)
This leads us to the following virtual moves:
We will use the term ``virtual moves'' to include the ordinary Reidemeister
moves as well as the four above moves; moves VVIII will be called
strictly virtual. Any two virtual knot diagrams connected by a
sequence of virtual moves (a virtual isotopy) represent the same
virtual knot.
Note that there is only one sort of potential move that we are not
allowing: we cannot move a strand with two real crossings over or under a
virtual crossing. These two moves are called The Forbidden Moves, which I'll
number IVt and IVh. The reason they are forbidden (excuse the pun :) is
that using both we can change any knot into any other knot, virtual or
classical! In terms of Gauss diagrams, the forbidden moves let us switch
the order of any two neighboring arrowheads (IVh) or arrowtails (IVt), so
that we can eventually end up with all the arrows trivializable by Type I
moves. Allowing both forbidden moves would render the theory of virtual
knots trivial, since every two virtual knot diagrams are equivalent by a
sequence of virtual and forbidden moves. Fortunately, we need not worry:
the forbidden move changes the Gauss code by reversing the order in which
we encounter two crossings, so it doesn't qualify as a move on virtual knot
diagrams which preserves the Gauss code and we have no reason to include
either forbidden move in our list of virtual moves.
In fact, here is a demonstration
of how to unknot the trefoil using the forbidden moves. We might
have trouble keeping our shoes on if we could use these moves!
Many of the usual theorems from classical knot theory carry over to virtual
knot theory with little or no modification; the Jones polynomial, the
quandle,
the fundamental group and others extend quite nicely. However, we often find
unexpected results, such as the existence of two distinct quandles associated
to each virtual knot and virtual knots which have trivial fundamental group
or trivial Jones polynomial but are not equivalent to the unknot.
Kauffman [1] has extended quantum link invariants, the quandle, the Jones
polynomial, and other
invariants of classical knots to virtual knots (in addition to discovering the
whole field of virtual knots in the first place), and a paper by Polyak,
Goussarov and Viro proposes a scheme for extending Vassiliev or finite
type invariants to virtual knots. This whole field is really rather new
(Kauffman announced it in
1996), and there is still a lot to be done.
References
 L. H. Kauffman,
Virtual knot theory. European J. Combin. 20 (1999)
663690.
 M. Goussarv, M. Polyak and O. Viro.
Finite type invariants
of classical and virtual knots. Topology 39 (2000) 10451068.
Copyright © 20002010 Sam Nelson
