Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in \(4\)-manifolds (Q6597295)

In this paper, the author introduces a family of knot concordance invariants \(\theta^{(q)}\) for each prime based on the equivariant Seiberg-Witten-Floer cohomology. These invariants \(\theta^{(q)}(K)\) are subadditive under the connected sum and provide sharper lower bounds for the slice genus \(g_4(K)\) of \(K\) than the knot signature. Moreover, \(\theta^{(q)}(K)\) give lower bounds for the genus of homologically trivial, smooth, properly embedded surfaces bounding \(K\) in a definite 4-manifold \(X\) with \(H_1(X)=0\) and \(\partial X = S^3\).\N\NThe invariants \(\theta^{(q)}\) are derived from the author's previous work [\textit{D. Baraglia} and \textit{P. Hekmati}, Algebr. Geom. Topol. 24, No. 1, 493--554 (2024; Zbl 1545.57013)]. There, the author constructed, for each prime \(q\), a sequence of integral-valued invariants for \(\mathbb{Z}_q\)- \(\operatorname{spin}^c\)-rational homology spheres \((Y,\mathfrak{s})\). The author denoted them by \(\delta_{\mathbb{Z}_2, Q^j}(Y,\mathfrak{s})\) for \(q=2\), and \(\delta_{\mathbb{Z}_q,S^j}(Y,\mathfrak{s})\) for \(q>2\). For simplicity, we use the same notation \(\delta_{\mathbb{Z}_q, j}(Y,\mathfrak{s})\) regardless of the parity of \(q\).\N\NFrom these results, one can define the knot concordance invariants \(\delta_j ^{(q)}(K):= 4 \delta_{\mathbb{Z}_q, j}(\Sigma_q(K), \mathfrak{s}_0)\), where \((\Sigma_q(K),\mathfrak{s}_0)\) is the \(q\)-fold cyclic cover of \(S^3\) branched along \(K\) equipped with the distinguished \(\mathbb{Z}_q\)-invariant \(\operatorname{spin}^c\)-structure \(\mathfrak{s}_0\). These invariants \(\delta^{(q)}_{j}(K)\) are decreasing as functions of \(j\ge 0\) and satisfy \(\delta^{(q)}_{j}(K)+\frac{1}{2}\sigma^{(q)}(K)\ge 0\). The equality holds for all \(j\ge g_4(K)-\frac{1}{2}\sigma(K)\) when \(q=2\) and all \(j\ge \frac{q-1}{2}g_4(K)-\frac{\sigma^{(q)}(K)}{4}\) when \(q>2\). Then define, for each integer \(m\ge0\), \N\[\N\begin{cases}\theta ^{(2)}(K,m) :=\operatorname{max}\left\{0, j^{(2)}(-K,m)-\sigma(K)/2\right\} & q=2\\\N\theta^{(q)}(K,m):=\operatorname{max}\left\{0, \frac{2 j^{(q)}(-K,m)}{q-1}-\frac{\sigma^{(q)}(K)}{2(q-1)}\right\} & q>2\end{cases}, \N\]\Nwhere \(j^{(q)}(-K,m)\) is the smallest integer \(j\ge 0\) such that \(\delta^{(q)} _{j}(-K)-\frac{1}{2}\sigma^{(q)}(K)=m\). Set \(\theta^{(q)}(K) = \theta^{(q)}(K,0)\). From the definition and the property of \(\delta_{\mathbb{Z}_q,j}\) discussed above, we have \(g_4(K)\ge \theta^{(q)}(K)\ge -\frac{\sigma^{(q)}(K)}{2(q-1)}\).\N\NNow consider a (negative) definite 4-manifold \(X\) with \(H_1(X)=0\) and \(\partial X = S^3\). Fix a characteristic \(c = (1,1,\cdots, 1)\in H_2(X)\) so that \(c^2 = - b_2(X)\). Given a knot \(K\subset \partial X= S^3\), let \(\Sigma\) be a properly embedded compact orientable surface \(\Sigma\) bounding \(K\). If \([\Sigma]\in H_2(X)\) is divisible by a prime \(q\), we form a \(q\)-fold cyclic branched cover \(\pi: W\to X\) of \(X\) along \(\Sigma\). Let \(\mathfrak{s}\) be a \(\mathbb{Z}_q\)-invariant \(\operatorname{spin}^c\)-structure of \(W\) satisfying \(\pi^*(c)=c_1(\mathfrak{s})\) and \(\mathfrak{s}|\partial W = \mathfrak{s}_0\). The author utilizes \(\delta\) and \(\delta_{\mathbb{Z}_q,j}\) invariants of \((W,\mathfrak{s})\) to deduce that \(g(\Sigma) \ge \theta^{(q)}(K,m)+\frac{q+1}{6q}[\Sigma]^2\) where \(m = -\frac{q^2-1}{6q} [\Sigma]^2\). For \(q=2\), define for \(x\in H_2(X)\), \(\eta(x)=\min\{(x+c)^2-b_2(X)\}\) where the minimum is taken over all characteristics \(c\) of \(X\). A similar argument as in the \(q>2\) case reveals that \(g(\Sigma) \ge \theta^{(2)}(K,m)+\frac{1}{4}[\Sigma]^2\) where \(m = -\frac{[\Sigma]^2}{4} + \eta([\Sigma]/2)\) for any characteristic \(c\). In both cases, we have \(g(\Sigma) \ge \theta^{(q)}(K)\) when \([\Sigma]=0\). In other words, \(g_H(K,X) \ge \theta^{(q)}(K)\).\N\NThe author also computes the \(\theta^{(q)}\)-invariants for various knots including some torus knots and the Whitehead double of torus knots.