If the back of your knee feels tight or swollen, it probably indicates the need to lower the saddle.

We all have different knees with different knee histories.

One fix does not fit all.

A generalization:

Fixed gear bikes do not cause knee problems as often as they reveal them.

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One sees a doctor, usually an orthopedic surgeon, in order to get an insurance-qualifying prescription for a sports physical therapist, orthotist (foot orthotics), or certified bicycle-fitter.
If one does not have insurance, it doesn't make much sense to see an orthopedic surgeon.

Saddle height:

With the pedal full down and the heel lowered (flat foot), the knee should still have a slight bend in it.

Fore-aft saddle position:

With the cranks horizontal, a plumbline hanging from the bump on the shin of the forward foot, just inside of the knee cap, should pass through the pedal spindle.
Just imagine you have no kneecap and hang the plumbline from the knee you would have without a kneecap, which puts the plumbline about an inch behind the kneecap.
Adjust the saddle fore and aft in order to move the plumbline fore and aft.

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A swollen back of the knee takes a long time to heal.

Alternating ice and heat, self-massage, elevation of the knee, baby aspirin, vitamin C and a lowered saddle.