Linking Number - Computing The Linking Number

Computing The Linking Number

There is an algorithm to compute the linking number of two curves from a link diagram. Label each crossing as positive or negative, according to the following rule:

The total number of positive crossings minus the total number of negative crossings is equal to twice the linking number. That is:

where n₁, n₂, n₃, n₄ represent the number of crossings of each of the four types. The two sums and are always equal, which leads to the following alternative formula

Note that involves only the undercrossings of the blue curve by the red, while involves only the overcrossings.