Seifert surfaces of dual knots
If you do surgery on a knot, you get a dual knot in the resulting manifold. The original knot and the dual knot have the same complement. A Seifert surface for the original knot is then a (rational) Seifert surface for the dual knot. We can see this dual knot and corresponding Seifert surface before we do the surgery.
In the surgered manifold, the green curve now bounds a meridional disk. Since the surgery was integral, the orange curve is now a longitude. Since it was a +4 surgery, the purple boundary of our Seifert surface runs 4 times longitudinally.
We got this copy of the dual by a pushoff isotopy, so drag the purple Seifert surface along. The original dual we got from surgery now intersects the surface transversally once. The surface intersects the surgery solid torus in a single meridional disk. I shrunk the surgery solid torus.
There’s a solid torus neighborhood of the surgery solid torus, the thickened dual knot and the parallelism of the dual knot to the orange meridian. Outside this, nothing has changed.