Linking number

3D-Link

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Linking Number -2

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Linking Number -1

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Linking Number 0

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Linking Number 1

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Linking Number 2

Linking Number is a fundamental concept in topology, a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. The linking number is a measure of the entanglement of two closed curves (or loops) in three-dimensional space. It is an integer that represents the number of times one loop winds around the other. The concept of linking number is crucial in understanding the topological properties of knots and links, and it has applications in various fields, including molecular biology, where it is used to describe the coiling of DNA molecules.

Definition

The linking number of two closed, non-intersecting curves in three-dimensional space can be defined in several equivalent ways. One common method is to project the curves onto a plane in such a way that the crossings are transverse (each crossing involves only two strands, one going over the other). At each crossing, assign a value of +1 or -1 depending on the orientation of the crossing. The linking number is then half the sum of these values over all crossings in the projection. This definition makes it clear that the linking number is a topological invariant, meaning it does not change under continuous deformations of the curves.

Mathematical Properties

The linking number has several important properties:

• It is symmetric: the linking number of curve A with curve B is the same as the linking number of curve B with curve A.
• It is an integer: although the calculation involves dividing the sum of crossing numbers by two, the result is always an integer.
• It is a topological invariant: as mentioned, the linking number does not change under continuous deformations of the curves (as long as the curves do not pass through each other).

Applications

In molecular biology, the linking number is used to describe the topology of DNA molecules. DNA double helices are often coiled upon themselves, forming structures known as supercoils. The linking number can help quantify the degree of coiling, which is important for understanding the physical properties of DNA and its behavior during processes such as replication and transcription.

In electromagnetism, the concept of linking number can be applied to magnetic field lines. The linking number of magnetic field lines can be related to the magnetic flux and has implications for the study of magnetic reconnection and the stability of plasma in fusion reactors.

See Also

• Knot theory
• Topological invariant
• DNA supercoiling
• Magnetic reconnection