Equivariant algebraic concordance of strongly invertible knots

Abstract

By considering a particular type of invariant Seifert surfaces we define a homomorphism $\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots $\stackrel{\sim }{C}$ to a new equivariant algebraic concordance group ${\stackrel{\sim }{G}}^Z$. We prove that $\Phi$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism (J. Lond. Math. Soc. (2023), no. 107, 2025-2053) and Alfieri and Boyle's equivariant signature (Michigan Math. J. 1 (2023), no. 1, 1–17). Moreover, we provide a partial result on the isomorphism type of ${\stackrel{\sim }{G}}^Z$ and 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 $\Phi$ can obstruct equivariant sliceness for knots with Alexander polynomial one.