From the article: >>> According to co-founder and CEO Alec Schkolnik, the X Engine design combines the high compression ratio and direct injection of a diesel engine with the constant volume combustion process of an Otto cycle engine and the over-expansion abilities of an Atkinson cycle engine, while solving the lubrication and sealing issues of the Wankel rotary engine and delivering huge power and efficiency
https://www.liquidpiston.com/how-it-works
1. air flows in as the chamber is expanding
2. fuel is injected when it is at maximum size
3. the chamber contracts to a tiny size to compress the fuel-air mix
4. combustion occurs, and runs to completion, in the tiny chamber
5. the chamber expands to extract work from the high-pressure gas
6. the chamber contracts to exhaust the waste gas,
The video makes the geometry clear. The way that the chamber contracts down to a tiny nub gives you the high compression ratio. The nubs being fixed in the stator lets you have injection. The fact that the combustion happens while the tip of the rotor is sliding over the nub gives you constant-volume combustion. I suppose the rotor being asymmetric lets you have overexpansion.