Concordance invariants from higher order covers

TL;DR

This paper extends the delta invariant to higher order covers, creating a family of invariants that form an infinite rank homomorphism from the smooth concordance group, revealing new distinctions among knots.

Contribution

It introduces a new family of concordance invariants from higher order covers, generalizing the delta invariant and demonstrating their infinite rank and independence from the signature.

Findings

The invariants form an infinite rank homomorphism from the concordance group.

They are generally not multiples of the knot signature.

Examples show these invariants distinguish knots beyond classical invariants.

Abstract

We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the direct sum of delta_{p^n} as n ranges through the natural numbers, yields a homomorphism of infinite rank from the smooth concordance group to Z^\infty. We also show that unlike delta, these new invariants typically are not multiples of the knot signature, even for alternating knots. A significant portion of the article is devoted to exploring examples.