One-Bipolar Topologically Slice Knots and Primary Decomposition

Abstract

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