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This is the construction / computation page for my joint paper with Roland van der Veen:

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 suggest that they should have a topological explanation. This we don't have yet.

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<< Mathematica Notebooks >>

 Notebook (.pdf)Source (.nb)CreatedLast ModifiedSummary
1 AAT source 2023-01-03 10:17:25 2023-01-04 02:16:05 $A$ - $A^T$ experiments.
2 AlexanderExperiments source 2022-09-22 13:33:05 2022-09-25 13:14:25 Alexander Experiments: Modifying the A Matrix.
3 APAI-Expand source 2022-09-13 19:29:10 2022-09-14 02:27:52 Same as APAI.nb, but with Expand replacing Factor.
4 APAILinks source 2023-11-09 05:43:29 2023-11-09 06:34:23 A version of $\rho_1$ for links with some testing, by Roland.
5 APAI - Other R source 2022-09-19 13:57:39 2022-09-19 14:50:56 Mathematica notebook for A Perturbed Alexander Invariant, with revised Rs.
6 APAI source 2022-01-08 07:25:07 2023-06-07 14:31:07 Mathematica notebook for A Perturbed Alexander Invariant.
7 ColouredA source 2023-08-04 01:28:50 2023-08-04 01:37:16 Just colour-coded $A$-matrices.
8 GGbar source 2023-01-13 04:23:20 2023-01-13 07:45:52 Failed $G$ - $\bar{G}$ experiments.
9 GGI source 2023-01-04 01:45:05 2023-01-04 02:31:12 Failed $G$ - $G-I$ experiments.
10 GGT source 2023-04-20 14:37:06 2023-04-20 14:58:25 Comparing $G$ and $G^T$.
11 GST48 source 2022-02-05 16:14:56 2022-02-05 17:02:17  
12 index source 2022-06-24 11:26:51 2022-06-24 11:50:30 This is the index file for the APAI project.
13 Invariance2 source 2022-02-17 10:15:55 2022-02-17 10:15:56 Invariance proof by Roland.
14 Invariance-Scratch source 2022-05-25 08:37:14 2022-06-02 15:39:47  
15 Lashings-230105 source 2023-03-02 18:21:44 2023-01-05 06:56:03 The Lashings Matrix.
16 Lashings source 2023-01-04 06:41:08 2023-04-20 13:34:40 The Lashings Matrix.
17 MakeData3-14 source 2022-02-14 08:32:49 2022-02-14 10:15:42 Compute and store ρ, Kh, and HOMFLYPT for knots with up to 14 crossings
18 Make source 2022-01-09 21:23:29 2023-06-07 15:27:18  
19 OtherAlexanders source 2022-01-23 17:14:22 2022-01-25 12:45:12 Other near-Wirtinger matrices to produce the Alexander polynomial.
20 PABI-InBack source 2022-01-07 13:06:43 2022-01-12 06:16:02 Perturbed Alexander-Burau Invariants. Continues Pensieve://2021-12/, but with the perturbation placed ahead of the bulk.
21 Palindromicity-GaussianLanguage-231219 source 2023-12-19 14:34:31 2023-12-19 14:34:47 Palindromicity by flipping and manipulating, in Gaussian integration language.
22 Palindromicity-GaussianLanguage-231220 source 2023-12-20 07:40:16 2023-12-20 05:57:40 Palindromicity by flipping and manipulating, in Gaussian integration language.
23 Palindromicity-GaussianLanguage source 2023-12-18 09:05:44 2023-12-22 06:27:44 Palindromicity by flipping and manipulating, in Gaussian integration language.
24 Palindromicity-MatrixLanguage source 2023-12-19 09:17:41 2023-12-20 04:29:57 Palindromicity by flipping and manipulating, in matrix language.
25 PerturbedGaussianIntegration-231222 source 2023-12-22 08:08:06 2023-12-22 07:48:34 Computing and playing with $\rho_1$ in the language of perturbed Gaussian Integration.
26 PerturbedGaussianIntegration-240103 source 2024-01-03 06:11:01 2024-01-03 06:10:17 Computing and playing with $\rho_1$ in the language of perturbed Gaussian Integration.
27 PerturbedGaussianIntegration source 2023-12-21 05:34:47 2024-01-03 10:33:52 Computing and playing with $\rho_1$ in the language of perturbed Gaussian Integration.
28 Playing-220818 source 2022-08-18 08:21:56 2022-08-18 08:40:19 Playing near $\rho_1$.
29 Playing-231222 source 2023-12-22 09:11:34 2023-12-22 09:14:29 The top row of $G$ isn't friendly.
30 Rho1-3-100 source 2022-01-06 11:05:01 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/.
31 Rot source 2022-05-24 09:23:57 2022-05-24 09:24:29 Computing rotation numbers. Based on Common.nb at pensieve://Classes/21-1350-KnotTheory/.
32 TimingGST48 source 2022-06-16 10:10:35 2022-06-17 04:42:04  

<< Links >>

  1. 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.
  2. Studied 2022-06-16 10:32:24: On the complexity of inverting integer and polynomial matrices - SpringerLink.
  3. Studied 2022-06-29 02:27:14: Car Crashes, Group Extensions and s-Cobordisms - Jerusalem Mathematics Colloquium.
  4. Studied 2023-10-23 09:22:16: Article Status.

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<< Other Files >>

AAT.m   abstract-dev.tex   abstract.tex   AlexanderExperiments.m   APAI4QT.aux   APAI4QT.tex   APAI4QT.zip   APAI.aux   APAI.brf   APAI-Dev.aux   APAI-Dev.brf   APAI-Dev.synctex   APAI-Dev.tex   APAI-Expand.m   APAILinks.m   APAI.m   APAINB.tex   APAI - Other R.m   APAI.tex   body.tex   ColouredA.m   defs.tex   ems-journal.sty   GGbar.m   GGI.m   GGT.m   gRules.tex   Invariance2.m   Invariance-R1l.tex   Invariance-R2c.tex   Invariance-R3-Short.tex   Invariance-R3.tex   Invariance-Rest.tex   Invariance-Scratch.m   Lashings.m   MakeData3-14.m   makefile   Make.m   new_aux   old_aux   OtherAlexanders.m   PABI-InBack.m   Palindromicity-GaussianLanguage.m   Palindromicity-MatrixLanguage.m   pd   PerturbedGaussianIntegration.m   picins.sty   Playing-220818.m   Playing-231222.m   refs4QT.tex   refs-dev.tex   refs.tex   Rot.m   TimingGST48.m  


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AAT.m   AAT.nb   abstract-dev.tex   abstract.tex   AlexanderExperiments.m   AlexanderExperiments.nb   APAI@230322.pdf   APAI4QT.aux   APAI4QT.pdf   APAI4QT.tex   APAI4QT.zip   APAI.aux   APAI.brf   APAI-Dev.aux   APAI-Dev.brf   APAI-Dev.pdf   APAI-Dev.synctex   APAI-Dev.tex   APAI-Expand.m   APAI-Expand.nb   APAI_Journal.pdf   APAILinks.m   APAILinks.nb   APAI.m   APAI.nb   APAINB.tex   APAI - Other R.m   APAI - Other R.nb   APAI.pdf   APAI.tex   Article Status.url   Blankeweer, Netherlands (1).jpg   Blankeweer, Netherlands (2).jpg   body.tex   Car Crashes, Group Extensions and s-Cobordisms - Jerusalem Mathematics Colloquium.url   ColouredA.m   ColouredA.nb   defs.tex   ems-journal.sty   Geneva-2206 - Shortcut.lnk   GGbar.m   GGbar.nb   GGI.m   GGI.nb   GGT.m   GGT.nb   gRules.tex   GST48.nb   G&T Decision.txt   icon.png   index.html   index.m   index.nb   Invariance2.m   Invariance2.nb   Invariance-R1l.tex   Invariance-R2c.tex   Invariance-R3-Short.tex   Invariance-R3.tex   Invariance-Rest.tex   Invariance-Scratch.m   Invariance-Scratch.nb   Lashings-230105.nb   Lashings.m   Lashings.nb   MakeData3-14.m   MakeData3-14.nb   makefile   Make.m   Make.nb   new_aux   Notebook.pdf   Oaxaca-2210 - Shortcut.lnk   old_aux   On the complexity of inverting integer and polynomial matrices - SpringerLink.url   OtherAlexanders.m   OtherAlexanders.nb   PABI-InBack.m   PABI-InBack.nb   Palindromicity-GaussianLanguage-231219.nb   Palindromicity-GaussianLanguage-231220.nb   Palindromicity-GaussianLanguage.m   Palindromicity-GaussianLanguage.nb   Palindromicity-MatrixLanguage.m   Palindromicity-MatrixLanguage.nb   pd   PerturbedGaussianIntegration-231222.nb   PerturbedGaussianIntegration-240103.nb   PerturbedGaussianIntegration.m   PerturbedGaussianIntegration.nb   picins.sty   Playing-220818.m   Playing-220818.nb   Playing-231222.m   Playing-231222.nb   [q-alg-9605023] Xiao-Song Lin, Feng Tian, Zhenghan Wang- Burau representation and random walk on string links.url   QT_Report_2.txt   QT_Report.txt   refs4QT.tex   refs-dev.tex   refs.tex   Rho1-3-100.nb   Rot.m   Rot.nb   SlideShow2.html   SlideShow.html   TimingGST48.m   TimingGST48.nb   VanDerVeen - Shortcut.lnk   Waco-2203 - Shortcut.lnk