This paper deals with a version of the equivariant slice problem. A knot \(K\) is strongly invertible if there is a (smooth) orientation preserving involution \(\tau:S^3\to S^3\) that fixes \(K\) set-wise and reverses the orientation of \(K\). The main subject of the current paper is the equivariant concordance group \(\widetilde{\mathcal{C}}\) which is defined by the collection of directed strongly invertible knots modulo equivariant concordance. The group law is given by the equivariant connected sum. Associated with \(\widetilde{\mathcal{C}}\) are the equivariant slice genus \(\widetilde{g}_4\) and the isotopy-equivariant slice genus \(\widetilde{ig}_4\). The invariant \(\widetilde{ig}_4(K, \tau)\) is defined to be the smallest possible genus of properly embedded orientable surfaces \(S\) in a smooth homology 4-ball \(W\) with a smooth diffeomorphism \(\tau_W:W\to W\) extending \(\tau\) such that \(\partial S= K \) and \(\tau_W(S)\) is smoothly isotopic rel \(\partial S\) to \(S\).
Given a strongly invertible directed knot \((K,\tau)\), the authors study the action of \(\tau\) on the knot Floer complex \(\mathcal{CFK}(K)\) and construct a homotopy involution \(\tau_K\) on \(\mathcal{CFK}(K)\) that enjoys some compatibility with the knot Floer involution \(\iota_K\) from [K. Hendricks and C. Manolescu, Duke Math. J. 166, No. 7, 1211–1299 (2017; Zbl 1383.57036)]. The knot Floer complex \(\mathcal{CFK}(K)\) together with this \(\tau_K\) and \(\iota_K\) forms a so-called \((\tau_K,\iota_K)\)-complex.
Abstractly, the authors define the group \(\mathfrak{R}_{\tau,\iota}\) consisting of abstract \((\tau_K, \iota_K)\)-complexes modulo local equivalence equipped with the group operation given by the tensor product. One of the main theorems of the paper states that the association \((K,\tau) \mapsto (\mathcal{CFK}(K),\tau_k, \iota_K)\) induces the group homomorphism \(\widetilde{\mathcal{C}} \to \mathfrak{R}_{\tau,\iota}\).
On the algebraic side \(\mathfrak{R}_{\tau,\iota}\), the authors extract numerical invariants \(\overline{V}^{\circ} _0\), and \(\underline{V}^{\circ} _0\), \(\circ\in \{\tau,\iota\tau\}\) resembling the involutive correction terms \(\overline{d}\) and \(\underline{d}\). Then they establish the inequality \(-\lceil \frac{1+\widetilde{ig}_4(K)}{2}\rceil \le \overline{V}^\circ _0 (K) \le \underline{V}^\circ _ 0 (K) \le \lceil \frac{1+\widetilde{ig}_4(K)}{2}\rceil\), \(\circ\in \{\tau,\iota\tau\}\). Given a strongly invertible slice knot \((K,\tau)\), the authors show that for any homology disk \(W\) with any extension \(\tau_W\) of \(\tau\), one has \(\max(\underline{V}^\tau _0 (K),\underline{V}^{\iota\tau}_0)\le V_0(\Sigma, \tau_W (\Sigma))\) for any slice disk \(\Sigma\) in \(W\), where \(V_0\) is the invariant defined in [A. Juhász and I. Zemke, “Stabilization distance bounds from link Floer homology”, Preprint, arXiv:1810.09158].
The authors also construct a family of strongly invertible slice knots \((K_n,\tau_n)\) such that \(\overline{V}^\tau _o (K_n)\ge n\) for all odd \(n\). This result, combined with the inequality in the previous paragraph, shows that \(\widetilde{ig}_4- g_4\) and \(\widetilde{g}_4 -g_4\) can be arbitrarily large, answering a question in [K. Boyle and A. Issa, J. Topol. 15, No. 3, 1635–1674 (2022; Zbl 1522.57008)].