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What device can rotate a knob and works with HA?
(feddit.it)
Home Assistant is open source home automation that puts local control and privacy first. Powered by a worldwide community of tinkerers and DIY enthusiasts. Perfect to run on a Raspberry Pi or a local server. Available for free at home-assistant.io
The issue is going to be that you need 3 full turns. It also sounds like you're not actually describing a "knob", meaning a normal doorknob, but some sort of metal wheel.
Picture may be more helpful, but it certainly sounds like nothing would exist for this.