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A Perturbed-Alexander Invariant.
Dedicated to the memory of V. F. R. Jones, 1952-2020, a friend and a mentor.
Paper PDF here: APAI.pdf. Computations here: APAI.nb.
Abstract. In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write one of the standard matrices $A$ whose determinant is its Alexander polynomial, yet instead of computing the determinant we consider a certain quadratic expression in the entries of $A^{-1}$. The proximity of our formulas to the Alexander polynomial suggests that they should have a topological explanation. This we do not have yet.
Archive Data figs Snips / Talks: Geneva-2206 Projects: HigherRank Talks: Oaxaca-2210 People: VanDerVeen Talks: Waco-2203 SlideShow2.html SlideShow.html / APAI@230322.pdf APAI@230608.pdf APAI4QT.pdf APAI-AsPublished.pdf APAI-Dev.pdf APAI_Journal.pdf APAI.pdf GalleyProof@.pdf GalleyProof.pdf Notebook.pdf / G&T Decision.txt QT_Report_2.txt QT_Report.txt<< Mathematica Notebooks >>
| Notebook (.pdf) | Source (.nb) | Created | Last Modified | Summary | |
|---|---|---|---|---|---|
| 1 | AAT | source | 2024-03-17 10:28:44 | 2023-01-04 02:16:05 | $A$ - $A^T$ experiments. |
| 2 | AlexanderExperiments | source | 2024-03-17 10:28:44 | 2022-09-25 13:14:25 | Alexander Experiments: Modifying the A Matrix. |
| 3 | APAI-Expand | source | 2024-03-17 10:28:44 | 2022-09-14 02:27:52 | Same as APAI.nb, but with Expand replacing Factor. |
| 4 | APAI-InvarianceForLinks | source | 2024-04-09 08:43:56 | 2024-04-09 08:47:54 | Mathematica notebook for A Perturbed Alexander Invariant. |
| 5 | APAILinks2 | source | 2024-04-01 15:49:53 | 2024-04-01 15:58:41 | APAI for links, using the adjugate matrix (by Roland). |
| 6 | APAILinks | source | 2024-03-17 10:28:44 | 2023-11-09 06:34:23 | A version of $\rho_1$ for links with some testing, by Roland. |
| 7 | APAI - Other R | source | 2024-03-17 10:28:44 | 2022-09-19 14:50:56 | Mathematica notebook for A Perturbed Alexander Invariant, with revised Rs. |
| 8 | APAI | source | 2024-03-17 10:28:44 | 2024-04-14 13:02:12 | Mathematica notebook for A Perturbed Alexander Invariant. |
| 9 | APAI-Positivity (1) | source | 2024-03-17 10:28:44 | 2024-03-05 17:00:43 | |
| 10 | APAI-Positivity | source | 2024-03-17 10:28:44 | 2024-03-05 07:14:59 | Roland checking Joe Boninger's conjecture that for a closure of a positive braid the coefficients of $\rho_1$ are all positive. Continued pensieve://Talks/Oaxaca-2210/Rho_d-Positivity.nb. |
| 11 | ColouredA | source | 2024-03-17 10:28:44 | 2023-08-04 01:37:16 | Just colour-coded $A$-matrices. |
| 12 | GGbar | source | 2024-03-17 10:28:44 | 2023-01-13 07:45:52 | Failed $G$ - $\bar{G}$ experiments. |
| 13 | GGI | source | 2024-03-17 10:28:44 | 2023-01-04 02:31:12 | Failed $G$ - $G-I$ experiments. |
| 14 | GGT | source | 2024-03-17 10:28:44 | 2023-04-20 14:58:25 | Comparing $G$ and $G^T$. |
| 15 | GST48 | source | 2024-03-17 10:28:44 | 2022-02-05 17:02:17 | |
| 16 | index | source | 2024-03-17 10:28:44 | 2022-06-24 11:50:30 | This is the index file for the APAI project. |
| 17 | Invariance2 | source | 2024-03-17 10:28:44 | 2022-02-17 10:15:56 | Invariance proof by Roland. |
| 18 | Invariance-Scratch | source | 2024-03-17 10:28:44 | 2022-06-02 15:39:47 | |
| 19 | Lashings-230105 | source | 2024-03-17 10:28:44 | 2023-01-05 06:56:03 | The Lashings Matrix. |
| 20 | Lashings | source | 2024-03-17 10:28:44 | 2023-04-20 13:34:40 | The Lashings Matrix. |
| 21 | MakeData3-14 | source | 2024-03-17 10:28:44 | 2022-02-14 10:15:42 | Compute and store ρ, Kh, and HOMFLYPT for knots with up to 14 crossings |
| 22 | Make | source | 2024-03-17 10:28:44 | 2024-04-14 13:02:42 | |
| 23 | OtherAlexanders | source | 2024-03-17 10:28:44 | 2022-01-25 12:45:12 | Other near-Wirtinger matrices to produce the Alexander polynomial. |
| 24 | PABI-InBack | source | 2024-03-17 10:28:44 | 2022-01-12 06:16:02 | Perturbed Alexander-Burau Invariants. Continues Pensieve://2021-12/, but with the perturbation placed ahead of the bulk. |
| 25 | Palindromicity-GaussianLanguage | source | 2024-03-17 10:28:44 | 2023-12-22 06:27:44 | Palindromicity by flipping and manipulating, in Gaussian integration language. |
| 26 | Palindromicity-MatrixLanguage | source | 2024-03-17 10:28:44 | 2023-12-20 04:29:57 | Palindromicity by flipping and manipulating, in matrix language. |
| 27 | PerturbedGaussianIntegration-Invariance | source | 2024-03-17 10:28:44 | 2024-02-28 12:20:05 | Invariance of $\rho_1$ in the language of perturbed Gaussian Integration. |
| 28 | PerturbedGaussianIntegration | source | 2024-03-17 10:28:44 | 2024-05-08 02:07:44 | Computing and playing with $\rho_1$ in the language of perturbed Gaussian Integration. |
| 29 | Playing-220818 | source | 2024-03-17 10:28:44 | 2022-08-18 08:40:19 | Playing near $\rho_1$. |
| 30 | Playing-231222 | source | 2024-03-17 10:28:44 | 2023-12-22 09:14:29 | The top row of $G$ isn't friendly. |
| 31 | Rho1-3-100 | source | 2024-03-17 10:28:44 | 2022-01-07 08:29:25 | Perturbed Alexander-Burau Invariants. DPG code from PerturbedHeisenberg2.nb at pensieve://Classes/21-1350-KnotTheory/. Continued pensieve://Projects/APerturbedAlexanderInvariant/. |
| 32 | Rho1NotLogConcave | source | 2024-03-26 12:51:50 | 2024-03-26 12:53:16 | With Joe Boninger: $\rho_1$ is not log-concave even on alternating knots. Continues pensieve://2023-03/ (for Alexander). |
| 33 | Rot | source | 2024-03-17 10:28:44 | 2022-05-24 09:24:29 | Computing rotation numbers. Based on Common.nb at pensieve://Classes/21-1350-KnotTheory/. |
| 34 | TimingGST48 | source | 2024-03-17 10:28:44 | 2022-06-17 04:42:04 |
<< Links >>
- Studied 2022-05-24 17:09:40: [q-alg-9605023] Xiao-Song Lin, Feng Tian, Zhenghan Wang- Burau representation and random walk on string links.
- Studied 2022-06-16 10:32:24: On the complexity of inverting integer and polynomial matrices - SpringerLink.
- Studied 2022-06-29 02:27:14: Car Crashes, Group Extensions and s-Cobordisms - Jerusalem Mathematics Colloquium.
- Studied 2023-10-23 09:22:16: Article Status.
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