Have a look at the this torus, embedded in a standard way in . The shaded disk is inside the torus with its boundary on the torus itself. Consider , the boundary of . It represents a nontrivial loop in the torus, but since it bounds a disk in , the loop is trivial in . Such a disk is called a **compressing disk** for the torus, and we can compress the torus by cutting along and gluing in two copies of as shown in the second picture. This is also called **surgering along** . In this case, we obtain a sphere by compressing along .

Notice that when we compress along we decrease the genus of our surface by one. With that in mind, consider the Seifert surface for the knot we looked at previously. Are there any compressing disks for ? Suppose there was a compressing disk and didn’t separate into two components . Then we could compress along , obtaining a new surface . Since we didn’t touch anything near the boundary of , is still a Seifert surface for . But is a punctured torus, so must be a disk, and the only knot bounding a disk is the unknot.

On the other hand, if there was a compressing disk such that **did** separate into two components, then one of these components would have to be an annulus. As above, we would get a disk bounding the knot . (Why? Working out these details is a fun exercise using Euler characteristic arguments.)

So if you believe that the trefoil is knotted, then the surface must be incompressible!

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