Alexander ideals of classical knots
Let \(k\) be a knot in \(S^3\) and let \(\widetilde{K}\) be the infinite cyclic cover of the closed complement of a tubular neighbourhood of \(k\). Then \(H_1(\widetilde{K})\) is a finitely generated module over the ring \(\Lambda={\mathbb{Z}}[t,t^{-1}]\). This module has associated with it a sequence, \(E_1\subset E_2\subset\cdots\subset E_n= E_{n+1}=\cdots= \Lambda\), of ideals of \(\Lambda\). The highest common factor of \(E_i\) is a Laurent polynomial \(\Delta_i(t)\), defined up to multiplication by \(\pm t^r\). We call \(E_i\) the \(i\)th Alexander ideal of \(k\) and \(\Delta_i(t)\) the \(i\)th Alexander polynomial of \(k\). It is known that a polynomial \(f(t)\in\Lambda\) arises as \(i\)-th Alexander polynomial of some knot if and only if \(f(1)= \pm 1\) and \(f(t^{-1})=\pm t^r f(t)\) for some \(r\) [\textit{J. Levine}, Topology 4, 135-141 (1965; Zbl 0134.43101); \textit{H. Seifert}, Math. Ann. 110, 571-592 (1934; Zbl 0010.13303)]. In this paper, the authors give a characterization of the ideals of \(\Lambda\) belonging to Alexander ideals of some knot as follows. Let \(A_i\) be the set of \(i\)th Alexander ideals of knots, and let \(A=\bigcup_{i=1}^{\infty}A_i\). They show that an ideal \(E\) of \(\Lambda\) belongs to \(A\) if and only if \(E=\overline{E}\) and \(\varepsilon(E)={\mathbb{Z}}\), where \(\varepsilon:\Lambda\rightarrow {\mathbb{Z}}\) is the ring homomorphism defined by \(\varepsilon(t)=1\) and \(\overline{E}\) is the ideal that is obtained from \(E\) by sending \(t\) to \(t^{-1}\). Combining this and the results of \textit{C. Kearton} [Bull. Am. Math. Soc. 79, 952-955 (1973; Zbl 0276.57006)] (or \textit{J. Levine} [Trans. Am. Math. Soc. 229, 1-50 (1977; Zbl 0653.57012)], or \textit{H. F. Trotter} [Knot Theory, Proc. Semin., Plans-sur-Bex 1977, Lect. Notes Math. 685, 291-299 (1978; Zbl 0407.57015)]), the authors remark that the set of Alexander ideals is the same for each set of simple \((2p-1)\)-knots, \(p=1,2,...\). The authors also show that \(A_1\subsetneq A_2\subset A_3\subset\cdots \subset A\). And they raise questions: Is it true that \(A_n\neq A_{n+1}\) for some \(n>1\)? Can one characterize \(A_n\)? Can any sequence be realized, subject to each \(E_i\) satisfying the two conditions known to be necessary? The corresponding question for Alexander polynomials has been answered affirmatively by \textit{J. Levine} [Topology 4, loc. cit.].
scientific article; zbMATH DE number 1146198