Satellite Operations and $θ$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914058867834880 |
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| author | McConkey, Rob Seaton, Luke J |
| author_facet | McConkey, Rob Seaton, Luke J |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Satellite Operations and $θ$ McConkey, Rob Seaton, Luke J Geometric Topology 57K14 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. |
| title | Satellite Operations and $θ$ |
| topic | Geometric Topology 57K14 |
| url | https://arxiv.org/abs/2509.22939 |