Linking in Cyclic Branched Covers and Satellite (non)-Homomorphisms

TL;DR

This paper provides an explicit formula for linking numbers in cyclic branched covers of knots and applies it to obstruct satellite operations from inducing homomorphisms on smooth concordance.

Contribution

It introduces a new explicit method for computing linking numbers in cyclic branched covers and uses this to analyze satellite operations in knot concordance.

Findings

Derived explicit linking number formulas for cyclic branched covers

Identified obstructions to satellite operations inducing concordance homomorphisms

Applied formulas to various cases demonstrating their utility

Abstract

Let $K\subset S^3$ be a knot and $\eta, \gamma \subset S^3\backslash K$ be simple closed curves. Denote by $\Sigma_q(K)$ the $q$-fold cyclic branched cover of $K$. We give an explicit formula for computing the linking numbers between lifts of $\eta$ and $\gamma$ to $\Sigma_q(K)$. As an application, we evaluate, in a variety of cases, an obstruction to satellite operations inducing homomorphisms on smooth concordance.