The knot Floer complex and the smooth concordance group
For a knot \(K\) in the \(3\)-sphere, the knot Floer complex \(CFK^\infty(K)\) was introduced by \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)] and \textit{J. Rasmussen} [Floer homology and knot complements, Ph. D. thesis, Harvard University, (2003), {\url arXiv:math/0306378}]. In a previous paper [J. Topol. 7, No. 2, 287-326 (2014; Zbl 1368.57002)], the author defined a new concordance invariant \(\varepsilon\) for knots \(K\). Indeed, \(\varepsilon\) is an invariant of the filtered chain homotopy type of \(CFK^\infty(K)\).NEWLINENEWLINEThe main purpose of the present paper under review is to give another concordance homomorphism from the smooth concordance group to a group \(\mathcal{F}\), which is obtained from the monoid of chain complexes \(\{CFK^\infty(K)\}\) by taking a certain quotient by using \(\varepsilon\).NEWLINENEWLINEThe point of \(\mathcal{F}\) is that it is totally ordered. By using this structure, a linear independence in the smooth concordance group can be detected. As examples, the topologically slice knots \(D_{p,p+1}\sharp - T_{p,p+1}\;(p\geq 1)\) are shown to be independent in the smooth concordance group, where \(D_{p,p+1}\) is the \((p,q)\)-cable of the positive untwisted double of the right-handed trefoil, and \(T_{p,p+1}\) is the \((p,q)\)-torus knot. Such families are known, but this is a new one.
scientific article; zbMATH DE number 6361412
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