08 April 2010
Is there an amphichiral 2-component link with linking number 8?
A 2-component link is two closed loops in 3-dimensional space, possibly tangled up with each other (linked with each other), and with each one possibly tied up in a knot. The image shows two 2-component links, one simple, one not so simple, with the two components shown as red and blue. A link is amphichiral if you can smoothly deform the link (no cutting, breaking, regluing, passing through itself allowed) in 3-dimensional space to make it look exactly like its mirror image. Clicking on the image and looking through the image gallery you will see that the simpler link shown is amphichiral. The linking number of a 2-component link is a number that you calculate by looking at a projection of the link onto a 2-dimensional surface and adding up +1's and -1's for all the crossings, according to a rule which is also illustrated in the image gallery. The linking number measures how linked the two different loops are with each other. For any integer n you can always find a 2-component link with linking number equal to n. But the easiest way to do those doesn't make an amphichiral link. The smallest value of n for which it is unknown whether or not there exists an amphichiral 2-component link with linking number n is n=8. Can you find an amphichiral 2-component link with linking number 8. Click on the image to see the definition of linking number carefully explained.
(Thanks to Paul Melvin for explaining this problem to us.)