Septic Arthritis: The patient's presentation with right knee pain, erythema, effusion, fever (100.7F), elevated leukocyte count (22,000), and synovial fluid analysis showing 45,000 microliters with 75% neutrophils strongly suggests septic arthritis. The absence of crystals and a negative gram stain does not rule out infection, as cultures may take time to become positive.

Gout: Although the synovial fluid analysis did not show crystals, gout can sometimes present without crystals in the fluid, especially if the patient is on anti-inflammatory medications or if the sample is not properly processed. The high neutrophil count could also be seen in gout, making it a consideration.
Pseudogout (Calcium Pyrophosphate Deposition Disease, CPPD): Similar to gout, pseudogout can cause significant inflammation and could potentially present with a high neutrophil count in the synovial fluid, although crystals would typically be expected.

Lyme Arthritis: In endemic areas, Lyme disease can cause arthritis that might mimic septic arthritis, especially if the patient has not noticed the characteristic rash. It's crucial to consider this diagnosis to avoid missing a treatable cause of arthritis.
Gonococcal Arthritis: This is a sexually transmitted infection that can cause septic arthritis, often presenting with dermatitis and tenosynovitis in addition to arthritis. It's a critical diagnosis not to miss due to its implications for contact tracing and treatment.
Tuberculous Arthritis: Although less common, tuberculous arthritis can present with chronic or subacute symptoms and might be considered in patients with risk factors for tuberculosis or those who do not respond to standard treatments for septic arthritis.

Rheumatoid Arthritis (RA) Flare: While RA can cause significant joint inflammation, the acute presentation with fever and such a high leukocyte count in the synovial fluid would be unusual. However, it's a consideration in patients with known RA.
Seronegative Spondyloarthropathies: Conditions like reactive arthritis, psoriatic arthritis, or ankylosing spondylitis can cause significant joint inflammation but typically have different clinical presentations and might not fit as well with the acute onset and laboratory findings described.
Hemarthrosis: Bleeding into the joint space, which could be due to trauma, anticoagulation, or a bleeding disorder, might present with joint effusion and could potentially cause an inflammatory response, although the fluid analysis would typically show blood rather than the described findings.