Satellite Operations and $\hat{I}¸$

Abstract

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

19 pages, 10 figures, .nb, .py, .csv