Satellite Operations and $θ$

We study the behavior of the knot invariant $θ$ under satellite operations. First, we prove that $θ$ is additive under connected sum. We then introduce a computational tool to generate t-twisted Whitehead doubles and apply it to explore the case of untwisted Whitehead doubles. We propose a conjecture describing the behavior of $θ$ on untwisted Whitehead doubles and verify the conjecture for the first 2977 prime knots. The pair of invariants $Θ= (Δ,θ)$ was introduced by Bar-Natan and van der Veen, where $Δ$ is the Alexander polynomial. The invariant $θ$ is easily computable and effective at distinguishing knots. Further exploration of satellite operations and $θ$ is proposed to reveal new patterns among cables and general satellites.