One-bipolar topologically slice knots and primary decomposition

Abstract

Let {T_n} 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 not equal to 1 the quotient 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 greater than 1 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 Ozsv\'ath-Szab\'o $d$-invariants, which is of independent interest. As another application, for each g greater than 0, we give a topologically slice knot of concordance genus g that is $\nu^+$-equivalent to the unknot.