The unknotting number and classical invariants. II. (Q2921064)

Let \(K\) be a knot in \(S^3\), and let \(B(K)=H_1(S^3\setminus{K};\mathbb{R}\Lambda)\), where \(\mathbb{R}\Lambda=\mathbb{R}[t,t^{-1}]\). This module has \(m\times{m}\) presentation matrices \(A\) which are hermitean (\(\overline{A}^{tr}=A\)) with respect to the involution \(t\mapsto{t^{-1}}\) of \(\mathbb{R}\Lambda\) and such that the Blanchfield pairing \(b_K:B(K)\times{B(K)}\to\mathbb{R}(t)/\mathbb{R}\Lambda\) is given by \(b_K([x],[y])=\overline{x}^{tr}Ay\), for all \(x,y\in\mathbb{R}\Lambda^m\). The minimal size \(m\) is a lower bound for the unknotting number for \(K\). Here this minimum \(n_\mathbb{R}(K)\) is computed in terms of Levine-Tristram signatures and nullities of associated matrices. Moreover, the minimum is realized by a diagonal matrix \(A\). Livingston's \(\rho\)-invariant for 4-ball genus may also be calculated in terms of Levine-Tristram signatures [\textit{Y. Ni}, J. Topol. 4, No. 4, 799--816 (2011; Zbl 1231.57014)], but in the final section it is shown that \(n_\mathbb{R}(K)\) and \(\rho(K)\) are independent invariants. (Remark. For \(R=\mathbb{Z}\), Proposition 2.1 predates the cited more general result of Ranicki. It was proven earlier by \textit{H. F. Trotter} [Lect. Notes Math. 685, 291--299 (1978; Zbl 0407.57015)].)