One-bipolar topologically slice knots and primary decomposition

Abstract

Let {Tn} be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran, Harvey, and Horn. It is known that for each n = 1 the group Tn/Tn+1 has infinite rank and T1/T2 has positive rank. In this paper, we show that T1/T2 also has infinite rank. Moreover, we prove that there exist infinitely many Alexander polynomials p(t) such that there exist infinitely many knots in T1 with Alexander polynomial p(t) whose nontrivial linear combinations are not concordant to any knot with Alexander polynomial coprime to p(t), even modulo T2. This extends the recent result of Cha on the primary decomposition of Tn/Tn+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.