A note on 0-bipolar knots of concordance order two

TL;DR

This paper investigates the structure of the smooth concordance group of topologically slice knots, demonstrating how certain knots can be used to understand the subgroup structure related to the bipolar filtration, particularly focusing on 0-bipolar knots of order two.

Contribution

It shows that a specific collection of knots used in previous work can be employed to analyze the quotient of the bipolar filtration subgroup, revealing new insights into the group's structure.

Findings

Identification of a collection of knots that generate a subgroup isomorphic to ^ in the quotient ^/.

Establishment of the relationship ^ < ^ in the bipolar filtration.

Extension of previous results to the context of 0-bipolar knots of order two.

Abstract

Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by Hedden, Kim, and Livingston to prove $\mathbb{Z}_2^{\infty} < \mathcal{T}$ can be used to see $\mathbb{Z}_2^{\infty} < \mathcal{T}_0/\mathcal{T}_1$.