Twisted Knots and the Perturbed Alexander Invariant
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911251365363712 |
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| author | Boninger, Joe |
| author_facet | Boninger, Joe |
| contents | The perturbed Alexander invariant $ρ_1$, defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of $ρ_1$ for families of knots $\{K_t\}$ given by performing $t$ full twists on a set of coherently oriented strands in a knot $K_0 \subset S^3$. We prove that as $t \to \infty$ the coefficients of $ρ_1$ grow asymptotically linearly, and we show how to compute this growth rate for any such family. As an application we give the first theorem on the ability of $ρ_1$ to distinguish knots in infinite families, and we conjecture that $ρ_1$ obstructs knot positivity via a "perturbed Conway invariant." Along the way we expand on a model of random walks on knot diagrams defined by Lin, Tian and Wang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted Knots and the Perturbed Alexander Invariant Boninger, Joe Geometric Topology Quantum Algebra The perturbed Alexander invariant $ρ_1$, defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of $ρ_1$ for families of knots $\{K_t\}$ given by performing $t$ full twists on a set of coherently oriented strands in a knot $K_0 \subset S^3$. We prove that as $t \to \infty$ the coefficients of $ρ_1$ grow asymptotically linearly, and we show how to compute this growth rate for any such family. As an application we give the first theorem on the ability of $ρ_1$ to distinguish knots in infinite families, and we conjecture that $ρ_1$ obstructs knot positivity via a "perturbed Conway invariant." Along the way we expand on a model of random walks on knot diagrams defined by Lin, Tian and Wang. |
| title | Twisted Knots and the Perturbed Alexander Invariant |
| topic | Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2403.03754 |