Linking Numbers

"NEW" LINK INVARIANTS

In this section we are going to introduce some link invariants that are very simple to calculate. These invariants were not used in the tabulation of knots, because they have the same value for all knots, and thus cannot distinguish among knots.

The first such invariant is the number of components. Clearly, if one link consists of x components and an other link component consists of y components, where x and y are not equal to each other, these two links cannot be equivalent.

While "visually" one can count the number of components very easily, we do need a method to obtain the number of components using only the notation. In "most" cases one gets this number by counting the number of "chains of neighbors", according to procedures discussed in previous sections. One however needs to be careful with links that have components with no crossings. Such links are not going to appear in the final table, since they are not "prime". But their invariants will have to be calculated, in order to show that the ones appearing in the tables, are not equivalent to these links.

Therefore, in order to ensure that we count the number of components correctly, we also have to take into account the "artificial" crossings. Each such crossing has to be accounted for two components (the ones it connects). If it can be accounted for only one component, this means that there exists an additional component with no real crossings.

If two links consist of the same number of components, and before one starts calculating more involved invariants, one may first resort to obtain the "linking numbers". These invariants were first considered by Gauss during the 19-th century, who explicitly obtained a formula to calculate them from their points.

The linking number of two link components is basically the "number of times" that each link "turns around" the other link. Mathematically speaking, the complement of a link component has fundamental group the group of integers, Z. The second component is actually a loop in such space, and therefore belongs to a particular homotopy class, characterized by an integer number. This number is actually the linking number. One may show that if one reverses the order of the two link components, the linking number does not change.

Since a linking number corresponds to a pair of distinct link components, if a link consists of n components, the total number of linking numbers is n (n-1) / 2. Therefore for n = 1 (knot), the number of linking numbers is 0, and this is why this concept cannot help in the problem of knots. Without resorting to the quite involved method of Gauss, here is a method of calculating the linking numbers.

Let two link components C1 and C2 "share" the (real) crossings (a1,a2), (b1,b2), (c1,c2), ... This means that for each such crossing, one crossing number belongs to C1 and one to C2. Each such crossing "contributes" either +1 or -1 towards the linking number. The sum of these "contributions", which (according to the Jordan Curve Theorem has to be an even number, is equal to twice the linking number.

The question remaining is how to determine the "sign" of each contribution. This turns out to be the "sign" of each crossing. One may recall that in order to check whether a "shadow" notation is drawable, we first had to obtain the "signs" of each crossing number. The "sign" of the entire pair can be defined as the "sign" of the overcrossing number.

If one switches the orientation of exactly one of the two link components, the linking number will be replaced by its opposite. Since we want to identify such links (they consist of exactly the same points), we are going to use as invariants not the actual linking numbers, but their absolute values.

Using this technique, one easily checks that the linking number of the two components of the Hopf link is equal to 1, and thus this link is non-trivial.

Next on our discussion is the issue of prime vs. the "other" links. One may read more about this problem by going to this page.


Charilaos Aneziris

Copyright 2001

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