A partial resolution of Hedden's conjecture on satellite homomorphisms

TL;DR

This paper investigates conditions under which satellite operations induce homomorphisms on the smooth knot concordance group, proving that certain patterns with specific winding numbers do not produce such homomorphisms, thus supporting Hedden's conjecture.

Contribution

It provides a partial proof that patterns with even winding numbers not divisible by 8 do not induce homomorphisms, advancing understanding of Hedden's conjecture.

Findings

Patterns with even winding numbers not divisible by 8 do not induce homomorphisms.

Supports Hedden's conjecture by identifying specific winding number conditions.

Advances the classification of satellite operations in knot concordance.

Abstract

A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden.