Linking number obstructions to satellite homomorphisms

TL;DR

This paper demonstrates that certain satellite operations in knot theory are not homomorphisms, using $d$-invariants of branched covers, and introduces a related technical result involving the Torelli group.

Contribution

It establishes new obstructions to satellite operations being homomorphisms and links $d$-invariants to the Torelli group, advancing understanding in knot concordance.

Findings

Satellite operations with specific positivity and winding number are not homomorphisms.

A new relation between $d$-invariants and the Torelli group is proved.

The results provide new tools for studying knot concordance and satellite operations.

Abstract

We prove that satellite operations that satisfy a certain positivity condition and have winding number other than one are not homomorphisms. The argument uses the $d$-invariants of branched covers. In the process, we prove a technical result relating $d$-invariants and the Torelli group which may be of independent interest.