Structure in the bipolar filtration of topologically slice knots

TL;DR

This paper investigates the structure of the bipolar filtration in the smooth concordance group of topologically slice knots, revealing infinite rank in the first quotient and analyzing invariants using d-invariants from branched covers.

Contribution

It demonstrates that the first quotient of the bipolar filtration has infinite rank and utilizes d-invariants from branched covers to establish key properties.

Findings

First quotient in bipolar filtration has infinite rank.

0-bipolar knots have vanishing tau, epsilon, and s invariants.

Results hold even modulo Alexander polynomial one knots.

Abstract

Let T denote the group of smooth concordance classes of topologically sice knots. We show that the first quotient in the bipolar filtration of T (i.e. 0-bipolar knots modulo 1-bipolar knots) has infinite rank, even modulo Alexander polynomial one knots. Any 0-bipolar knot has vanishing tau-, epsilon-, and s-invariants. We prove the result using d-invariants associated to the 2-fold branched covers of knot complements.