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

Abstract

We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, 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 $\theta^{(2)}(K)$ shares many similarities with the knot Floer homology invariant $\nu^+(K)$ defined by Hom and Wu. Our invariants $\theta^{(q)}(K)$ give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding $K$ in a definite $4$-manifold with boundary $S^3$.