Orders of knots in the algebraic knot cobordism group
The algebraic knot cobordism group \(G_{\pm}\) was introduced by \textit{J. Levine} [Comment. Math. Helv. 44, 229-244 (1969; Zbl 0176.221)] in order to study the cobordism groups of codimension two knots. In Invent. Math. 8, 98-110 (1969; Zbl 0179.524), he gave a complete set of invariants for \(G_{\pm}\) and showed that \(G_{\pm}\) is isomorphic to \({\mathbb{Z}}^{\infty}\oplus ({\mathbb{Z}}/2{\mathbb{Z}})^{\infty}\oplus ({\mathbb{Z}}/4{\mathbb{Z}})^{\infty}\). In particular the order a(K) of an odd dimensional knot K in the algebraic knot cobordism group is equal to 1, 2, 4 or infinite, and it is determined as follows. Theorem A. (1) a(K) is finite if and only if the local signature \(\sigma_{\phi}(K)\) vanishes for every symmetric irreducible real factor \(\phi\) (t) of the Alexander polynomial \(\Delta\) (t) of K. (2) Suppose that a(K) is finite. Then \(a(K)=4\) if and only if for some p-adic number field \({\mathbb{Q}}_ p\), there exists a symmetric irreducible factor \(\lambda\) (t) of \(\Delta\) (t) over \({\mathbb{Q}}_ p\) such that \[ ((-1)^ d\lambda (1)\lambda (-1),-1)_ p=-1\text{ and } {\mathcal E}_{\lambda}(K)=1. \] Here (,)\({}_ p\) is the Hilbert symbol and \(d=(1/2)\deg \lambda (t)\), and \({\mathcal E}_{\lambda}(K)\) is defined as follows. Let \(\Phi\) (t) be the symmetric irreducible factor of \(\Delta\) (t) over \({\mathbb{Q}}\) which has \(\lambda\) (t) as an irreducible factor over \({\mathbb{Q}}_ p\). Then \({\mathcal E}_{\lambda}(K)\) is the exponent of \(\Phi\) (t) in \(\Delta\) (t) modulo 2. However, in order to determine whether \(a(K)=4\) or not, we must check the Hilbert symbols for every prime number. The purpose of this paper is to prove the following theorem, which improves Theorem A and enables us to determine a(K) through a finite procedure. Theorem. if \(p\nmid 2\Delta (-1)\), then \(((-1)^ d\lambda (1)\lambda (- 1),-1)_ p=+1\) for any symmetric irreducible factor \(\lambda\) (t) of \(\Delta\) (t) over \({\mathbb{Q}}_ p.\) Thus, to determine whether \(a(K)=4\) or not, it suffices to check the Hilbert symbols only for prime factors of \(2\Delta (-1).\) By using this theorem, we determine a(K) of every prime classical knot K up to 10- crossings.
cobordism groups of codimension two knots
scientific article; zbMATH DE number 4155465