One-Bipolar Topologically Slice Knots and Primary Decomposition

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+1has infinite rank and T ₁/T ₂has positive rank. In this paper, we show that T ₁/T ₂also has infinite rank. Moreover, we prove that there exist infinitely many Alexander polynomials p(t) such that there exist infinitely many knots in T₁with 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+1for 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.