Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants ${𝜃}^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree $q$ cyclic cover of $S^3$ branched over $K$. In the case $q=2$, our invariant ${𝜃}^{(2)}(K)$ shares many similarities with the knot Floer homology invariant ${\nu }^{+}(K)$ defined by Hom and Wu. Our invariants ${𝜃}^{(q)}(K)$ give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding $K$ in a definite $4$-manifold with boundary $S^3$.