We establish a version of Seiberg-Witten Floer K-theory for knots, as well as a version of Seiberg-Witten Floer K-theory for 3-manifolds with involutions. The main theorem is a 10/8-type inequality for spin 4-manifolds with boundary and with involutions, and that yields numerous applications to knots, such as genus bounds and stabilizing numbers. We also give applications to get obstructions to extending involutions on 3-manifolds to spin 4-manifolds.
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