The Linking Number

A link

A link is defined as two closed curves in three-dimensional space. The first picture shows an example of a link with one yellow curve and one blue. The linking number is a simple numerical invariant of a link. Intuitively, it represents the number of times that each curve winds around the other. For example, if it is possible to pull the two curves apart, the linking number is zero.

When I studied the linking number, I would look at a picture of a link trying to calculate this number. It was confusing. It only became easy after I started crocheting. For example, the second picture shows the same link as the first but slightly rearranged. I simply slid the yellow loop along the blue one until I could clearly see a piece of the blue loop as a straight segment and the yellow loop circling around it. Now, it is easy to see that the yellow loop winds around the blue one 3 times, making the linking number 3.

The only thing to remember is that while counting the number of windings, I need to consider the direction. It is possible for a loop to wind clockwise and then counterclockwise. In this case, the linking number is the difference between the two.

I crocheted a lot of links, and now my students and I have no problem calculating the linking numbers.

The linking number


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