# Definition: Knot Diagram, Classical Crossing, Virtual Crossing

According to Peter Guthrie Tait (1831 - 1901), a knot diagram is a projection of a closed curve in three-dimensional space $$\mathbb R^3$$ onto a plane. For instance, projecting the following curve onto a plane (image source: bookofproofs)

could result in the following knot diagram: (image source: bookofproofs)

Every knot diagram fulfills the following three properties:

1. The projected curve is smooth, i.e. there are no cusps and every point has a tangent line.
2. Points of intersection are formed by at most two segments of the curve. These points are called double points. In the above example, the points $$A, B, C$$ and $$D$$ are double points.
3. Every double point has exactly two different tangent lines (this eliminates the possibility of a double point with only one tangent line, where the curve does not actually cross itself but where two line segments still touch each other at a point).
4. Each curve can be approximated by a finite number of line segments.

There are two types of double points:

• A classical crossing is a double point in a plane marked with under/overpassing information of the underlying three dimensional curve. In the above example, the double points $$A,B$$ and $$C$$ are classical crossings.
• A virtual crossing is a double point in a plane, which has no corresponding under/overpassing three dimensional curve. Rather, the idea is that the virtual crossing is not really there. In the above example, the point $$D$$ is a virtual crossing. By convention, a virtual crossing is denoted by a small circle around it.

Definitions: 1 2
Proofs: 3
Propositions: 4

Github: ### References

#### Bibliography

1. Dye, Heather: "An Invitation to Knot Theory", CRC Press, 2016
2. Kauffman, L.: Virtual Knot Theory