Satellite Operations and $\theta$

Rob McConkey; Luke J Seaton

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arXiv · 2509.22939

Satellite Operations and $\theta$

Rob McConkey·Luke J Seaton

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Abstract

We study the behavior of the knot invariant $\theta$ under satellite operations. First, we prove that $\theta$ 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 $\theta$ on untwisted Whitehead doubles and verify the conjecture for the first 2977 prime knots. The pair of invariants $\Theta = (\Delta,\theta)$ was introduced by Bar-Natan and van der Veen, where $\Delta$ is the Alexander polynomial. The invariant $\theta$ is easily computable and effective at distinguishing knots. Further exploration of satellite operations and $\theta$ is proposed to reveal new patterns among cables and general satellites.

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BibTeX RIS

Rob McConkey, Luke J Seaton. 2025-09-26. Satellite Operations and $\theta$. https://arxiv.org/abs/2509.22939

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