This is a tough question to answer due to the fact that most of the causes of white lines around the knees are from some form of stretch marks around the knees. What most people do not realize is that stretch marks are not only caused from rapid weight gain and weight loss, there are a number of other causes of these symptoms that could also lead to stretch marks appearing on the skin.
To start off, many people suffering from joint swelling will experience rapid swelling of the joints. The knee is one of the joints most commonly associated with this type of inflammation. This can be observed in conditions like rheumatoid arthritis or trauma to the knee. This trauma can occur from strenuous exercise, spending a lot of time on your knees for job requirements or it could even be idiopathic. You will not have any obvious weight gain during this period of time but your knee can become slightly bigger. This may not be something that you notice yourself because our bodies are able to adjust to small changes without you being aware of what is happening. The only evidence that could be left behind would be these white markings on your legs. Hydrating yourself and the topical application of certain moisturizers can also help expedite the process of getting past the markings.
Another potential cause of white lines around your knees could be as a side effect to medication. Medications like steroids can put a lot of strain on your skin as a side effect of the drug. The skin will actually become a little thinner when using this type of medication and any dermal abrasions could be more easily seen on the skin as a result. Because our knees are some of the most important joints in our body and have to support the weight of our entire frame. This stress can cause the skin to crease slightly as it is more fragile than normal and lead to white lines appearing on the skin. Thankfully, these lines will resolve on their own in just a matter of time and in most applications, steroids will not be used long-term due to their potential side effects so once you stop the medication, you will start to notice the marks fading.
You may also consider having a consultation with a dermatologist to rule out any possible skin infections that could lead to these symptoms. A common skin condition called alopecia areata is capable of creating white patches on your skin. Although these are usually found around your face, it is possible to have the disease appear only around your knees. This will also be associated with hair loss but because the knee is an area already with a small amount of hair, the hair loss may not be noticeable. This is a disease that can arise during times of problems with the immune system so make sure you are in a good state of health with a consultation to your local doctor. 
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