One-bipolar topologically slice knots and primary decomposition

TL;DR

This paper extends the understanding of the bipolar filtration of topologically slice knots, proving that the first level has infinite rank and exploring the primary decomposition related to Alexander polynomials.

Contribution

It demonstrates that T_1/T_2 has infinite rank and constructs knots with specific Alexander polynomials whose linear combinations are not concordant to knots with coprime polynomials.

Findings

T_1/T_2 has infinite rank.

Existence of knots with specific Alexander polynomials not concordant to coprime polynomials.

Surgery manifolds of satellite links have the same Ozsváth-Szabó d-invariants.

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.