Knot Diagrams

Let $K=f(\mathbb{S}^1) \subset \mathbb{R}^3$ be a knot, where $f: \mathbb{S}^1 \rightarrow \mathbb{R}^3$. Let $P \subset \mathbb{R}^3$ be a plane such that $K \cap P = \emptyset$. Let $p: \mathbb{R}^3 \rightarrow P$ be the orthogonal projection. Then we may assume without loss of generality that the restriction $p\circ f : \mathbb{S}^1 \rightarrow P$ is a generic immersion. This implies that the image $(p \circ f)(\mathbb{S}^1)$ is an immersed closed curve only with transverse double intersection points (called crossings or crossing points), other than embedded points.

Figure: Knot diagrams

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 $(p \circ f)(\mathbb{S}^1)$ with under-arc broken to indicate the crossing information. Here, a direction orthogonal to $P$ is chosen as a height direction, and if a crossing point is formed by $p(\alpha)$ and $p(\beta)$ where $\alpha$, $\beta$ are arcs in $K$, and $\alpha$ is located in higher position than $\beta$ is with respect to the fixed height direction, then $p(\alpha)$ is called an over-arc (or upper-arc) and $\beta$ 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 $1, \ldots, k$ in this order to the crossings that are encountered, where $k$ 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 $123123$ (Fig.  (1)).

To represent over- and under-crossing, if the number $j$ corresponds to an over- or under-crossing, respectively, then replace $j$ by $j_O$ or $j_U$, respectively. Thus the right-hand trefoil has the code $1_O 2_U 3_O 1_U 2_O 3_U$ (Fig.  (2)). To represent the sign of the crossing, replace them further by $j_{C +}$ or $j_{C -}$ if $j$ corresponds to a positive or negative crossing, respectively, where $C=O$ or $U$ 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 $1_{O+} 2_{U+} 3_{O+} 1_{U+} 2_{O+} 3_{U+}$. When one of the signs is reversed, a diagram no longer is realized on the plane as depicted in Fig.  (4) for $1_{O+} 2_{U+} 3_{O-} 1_{U+} 2_{O+} 3_{U-}$ (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.

Figure: Gauss codes