Concordance and mutation. (English) Zbl 1002.57007

Kirk, P.; Livingston, C.

Geom. Topol. 5, 831-883 (2001).

In 1961, Fox has defined the knot concordance group \(\mathcal{C}\). The group \(\mathcal{C}\) is abelian and countable. By Levine’s work in 1969, there is an epimorphism \(\phi: \mathcal{C} \to G = {\mathbf Z}^{\infty} \oplus {\mathbf Z}_2^{\infty} \oplus {\mathbf Z}_4^{\infty}\), so that \(\mathcal{C} / \text{Ker}(\phi) \cong G\). In the article under review, the authors recall results about the group \(\mathcal{C}\) obtained by Fox, Milnor, Casson, Gordon, Jiang, Freedman, Donaldson, Witten, Cochran, Orr, Teichner, and Livingston. In particular, \(\text{Ker}(\phi)\) has a subgroup isomorphic to \({\mathbf Z}^{\infty}\) and a subgroup isomorphic to \({\mathbf Z}_2^{\infty}\). However, as the authors emphasize, the structure of the group \(\mathcal{C}\) remains a mystery. The authors consider a new group \(\mathcal{M}\), the Concordance Group of Mutants, defined as the quotient of \(\mathcal{C}\) obtained by using the equivalence relation generated by positive mutation. The authors show that positive mutation preserves the \(S\)-equivalence class of a knot, and using this fact, they show that Levine’s epimorphism \(\phi: \mathcal{C} \to G\) factors as \(\phi_2 \circ \phi_1\), where \(\phi_1: \mathcal{C} \to \mathcal{M}\) and \(\phi_2: \mathcal{M} \to G\). The main result of the article asserts that the kernels of \(\phi_1\) and \(\phi_2\) are infinitely generated, containing subgroups isomorphic to \({\mathbf Z}^{\infty}\), and the kernel of \(\phi_2\) contains a subgroup isomorphic to \({\mathbf Z}_2^{\infty}\). In order to obtain the result, the authors construct families of knots which generate appropriate subgroups of \(\text{Ker}(\phi_1)\) and \(\text{Ker}(\phi_2)\). The main applied technique in proving the result is the use of Casson–Gordon invariants. Before the article under review was published, only one example of distinct (up to concordance) positive mutants has been known. Among other constructions, the authors of the article present for the first time infinite families of knots that are distinct (up to concordance) from their positive mutants.

Reviewer: Krzystof Pawalowski (Poznań)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57M27	Invariants of knots and \(3\)-manifolds (MSC2010)

Keywords:

Knot theory; concordance; mutation

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