Linking number

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In mathematics, the linking number is a simple invariant for links (i.e. submanifolds of three-dimensional space homeomorphic to a number of circles). It is initially defined for two-component links, but can easily be extended to deal with one-component links, i.e. knots.

The linking number of a link is calculated as follows:

Choose a direction on each component.

Draw a diagram of the link.

Look at the crossings of the components.

For each crossing, note the directions at the crossing point.

As in the picture, the crossings may come in either of the two forms:

(+)		(−)

(You might need to tilt your diagram until you get one of these pictures).

(The decision which type on crossing will be regarded as positive and which negative is an arbitrary convention).

Add the plusses and minuses for all the crossings. You are guaranteed to get an even integer. Its half is the linking number.

[edit] The linking number of a knot

A knot is a 1-component link. The aforementioned method for calculating a linking number disregards any twists and knots within single link component, so for a knot you always get zero as the linking number.

In general a simple count of crossings will not give a knot invariant - just consider an unknot diagram with one crossing. We can only get invariants up to regular isotopy. One such is Kauffman's self-linking number, the linking number of the knot K with a knot obtained by moving K up slightly. Whitney index is the total winding number of the tangent vector of knot diagram considered as a plane curve and is another regular isotopy invariant.

Alternatively one can get an invariant if one supplies a framing, i.e. a choice of a non-tangent vector to the knot at each point. Every knot has a Seifert surface (not unique!). Given a framing you can push the knot slightly in the direction of the framing vector. The resulting push-off will intersect the Seifert surface in a number of points. Each intersection point will have a sign coming from the orientation of the knot and the Seifert surface. Counting those points with signs will give an invariant of the framed knot, independent of the choice of the Seifert surface. This invariant is the self-linking number of the framed knot.

The Seifert surface itself gives a framing to the knot - just take the vector pointing into the surface. This framing is called the Seifert framing and gives self-linking number zero. The Kauffman self-linking number is just the self-linking number given by the "up" framing, also known as the blackboard framing.

[edit] See also

• Linking coefficient (for a more analytic / topological discussion of the same concept)