Equivariant algebraic concordance of strongly invertible knots

By considering a particular type of invariant Seifert surfaces we define a homomorphism $\mathrm{\Phi }$ from the (topological) equivariant concordance group of directed strongly invertible knots $\tilde{\mathcal{C}}$ to a new equivariant algebraic concordance group ${\tilde{\mathcal{G}}}^{\mathbb{Z}}$. We prove that $\mathrm{\Phi }$ lifts both Miller and Powell’s equivariant algebraic concordance homomorphism, and Alfieri and Boyle’s equivariant signature. Moreover, we provide a partial result on the isomorphism type of ${\tilde{\mathcal{G}}}^{\mathbb{Z}}$, and we obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox-Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that $\mathrm{\Phi }$ can obstruct equivariant sliceness for knots with Alexander polynomial one.