Let
be a knot,
where
.
Let
be a plane such that
.
Let
be the orthogonal projection.
Then we may assume without loss of generality that
the restriction
is a generic immersion.
This implies that the image
is an immersed
closed curve only with transverse double intersection points
(called crossings or crossing points),
other than embedded points.
The situations excluded under this assumption are points of tangency, triple intersection points, and points where a tangent line does not exist, for example. A reason why we may assume such a nice situation is that such projection directions are open and dense among all possible directions.
A knot diagram is such a projection image
with under-arc broken to indicate the crossing information.
Here, a direction orthogonal to
is chosen as a height direction,
and if a crossing point is formed by
and
where
,
are arcs in
, and
is located in
higher position than
is with respect to the fixed height direction,
then
is called an over-arc (or upper-arc)
and
a under-arc (or lower-arc).
The preimages of the crossing points in these arcs are called
over-crossing and under-crossing, respectively.
This information as to which arc is upper and which is lower, is
the crossing information.
It may be convenient to take the height function to be pointing toward the viewer
of a knot, see Fig.
.
To represent knot projections and diagrams symbolically,
Gauss codes [Gauss1883]
has been used. Such codes have been used to
produce knot tables [HTW98], and compute knot invariants by computer.
There are several conventions for such codes.
Here we follow the conventions in [Kau99].
We start with codes for projections. Pick a base point on a projection, and
travel along the curve in the given orientation direction.
Assign positive integers
in this order to the crossings that are encountered,
where
is the number of crossings. Then the Gauss code of the projection is
a sequence of integers read when the whole circle is traced back to the base point.
A typical projection of a trefoil has the code
(Fig.
(1)).
To represent over- and under-crossing,
if the number corresponds to an over- or under-crossing, respectively,
then replace
by
or
, respectively.
Thus the right-hand trefoil has the code
(Fig.
(2)).
To represent the sign of the crossing, replace them further by
or
if
corresponds to a positive
or negative crossing, respectively,
where
or
depending on over or under.
The convention for the sign of a crossing is depicted in Fig.
(3).
The signed code for the right-hand trefoil is
.
When one of the signs is reversed, a diagram no longer
is realized on the plane as depicted in
Fig.
(4) for
(this example is taken from [Kau99]).
In this figure a non-existent crossing, called a virtual crossing,
is represented by a circled crossing.
Such a generalization of knot diagrams and non-planar Gauss codes
up to equivalence by analogues of Reidemeister moves
is called virtual knot theory [Kau99].
These symbolic representations are useful in applications to DNA.