The Earth's climate system is a complex system that includes key components such as the Arctic Summer Sea Ice or the El Niño Southern Oscillation as well as climate tipping elements like the continental-scale ice sheets or the Amazon rainforest. Crossing warming thresholds of these elements can lead to a qualitatively different climate state, endangering the stability of human societies. Particularly, the cryosphere elements are vulnerable at current levels of global warming (1.2°C) while also having long response times and large structural uncertainties. Investigating a network of interacting Earth system components using an established conceptual model, we systematically assess which uncertainties of key Earth system component have the largest impacts on tipping risks. We find that the cryosphere tipping elements (the Greenland and the West Antarctica ice sheets) are most decisive for tipping risks and cascading effects within our model. At a global warming level of 1.5°C, neglecting the large cryosphere tipping elements can reduce the mean number of disintegrated Earth system components by as much as 56%. This is concerning as overshooting 1.5°C of global warming is fast becoming inevitable, while current state-of-the-art IPCC-type global circulation models do not (yet) include dynamic ice sheets. Our results suggest that urgent integrated Earth system model development and Earth observation efforts including the large polar ice sheets are necessary and a precautionary measure of meeting stringent climate targets is crucial to limit tipping risks.