## Abstract

Let {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 _= 1 the group T _{n}/T _{n+1}has infinite rank and T _{1}/T _{2}has positive rank. In this paper, we show that T _{1}/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 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 T 2. This extends the recent result of Cha on the primary decomposition of T _{n}/T _{n+1}for all n ≥ 2 to the case n = 1. To prove our theorem, we show that the surgery manifolds of satellite links of ν +- 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 ≥ 1, we give a topologically slice knot of concordance genus g that is ν +-equivalent to the unknot.

Original language | English |
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Pages (from-to) | 8314-8346 |

Number of pages | 33 |

Journal | International Mathematics Research Notices |

Volume | 2022 |

Issue number | 11 |

DOIs | |

State | Published - 1 Jun 2022 |

### Bibliographical note

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