Carbon Cycle
The snow leopard has the same role in the carbon cycle as all living things. Their job is to do cellular respiration to produce carbon dioxide, water, and Energy in the form of ATP. Also when they die, they decay, which also contributes to they carbon cycle. One last way they tie into the carbon cycle is when they produce waste products, the waste yeilds fossil fuels.
Water Cycle
The snow leopard, Like any other animal, fits into the water cycle for two reasons. All animals take in water from the environment when they drink. Two, when animals urinate, they are realising the water they have taken in, back into the environment to be evaporated. This is a continuous cycle within the water cycle.
Nitrogen Cycle
When the snow leopard produces organic mater it goes into the soil witch leads to mineralization, which yields ammonium, then nitrification happens which makes nitrites, and then with further nitrification it yields nitrates, which leads to more plant consumption, resulting in more organic mater. This is just part of the nitrogen cycle, but with ought the snow leopard or other animals, the cycle cannot start.