One-Bipolar Topologically Slice Knots and Primary Decomposition

Abstract Let $\{{\mathcal{T}}_n\}$ be the bipolar filtration of smooth concordance group topologically slice knots, which was introduced by Cochran et al. It is known that for each $n\ne 1$ ${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$ has infinite rank and ${\mathcal{T}}_1/{\mathcal{T}}_2$ positive rank. In this paper, we show also Moreover, prove there exist infinitely many Alexander polynomials $p(t)$ such knots in ${\mathcal{T}}_1$ with polynomial whose nontrivial linear combinations are not concordant to any knot coprime $p(t)$, even modulo ${\mathcal{T}}_2$. This extends recent result Cha on primary decomposition all $n\ge 2$ case $n=1$. To our theorem, surgery manifolds satellite links $\nu ^+$-equivalent same pattern link have Ozsváth–Szabó $d$-invariants, independent interest. As another application, $g\ge 1$, give a genus $g$ unknot.