2-torsion in the n-solvable filtration of the knot concordance group

TL;DR

This paper demonstrates that for the n-solvable filtration of the knot concordance group, the associated graded groups contain infinitely many elements of order 2, with specific properties and distinctions based on classical invariants.

Contribution

It proves the existence of infinitely many 2-torsion elements in the graded groups of the n-solvable filtration, expanding prior knowledge from lower levels to higher levels.

Findings

Each G_n contains infinite 2-torsion elements for n>1.

These elements are represented by negative amphichiral knots with trivial classical invariants.

The knots are distinguished by their Alexander polynomials and higher-order Alexander modules.

Abstract

In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order 2 (this was known previously for n=0,1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, tau-invariant, delta-invariants and Casson-Gordon invariants. Moreover each is smoothly slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in G_n, distinguished by their classical Alexander polynomials and by the orders of elements in their higher-order Alexander modules.