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.
McConkey et al. (2025) studied this question.