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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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Archive   Data   figs   Snips   /   Talks: Geneva-2206   Talks: Waco-2203  

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 Notebook (.pdf)Source (.nb)CreatedLast ModifiedSummary
1 AlexanderExperiments source 2022-09-22 14:33:05 2022-09-22 14:43:38 Alexander Experiments: Modifying the A Matrix.
2 APAI-Expand source 2022-09-13 20:29:10 2022-09-14 03:27:52 Same as APAI.nb, but with Expand replacing Factor.
3 APAI - Other R source 2022-09-19 14:57:39 2022-09-19 15:50:56 Mathematica notebook for A Perturbed Alexander Invariant, with revised Rs.
4 APAI source 2022-01-08 08:25:07 2022-06-24 09:11:42 Mathematica notebook for A Perturbed Alexander Invariant.
5 Experiments source 2022-02-01 13:59:53 2022-02-01 14:00:33 A Perturbed Alexander Invariant, with formulas from PABI-InBack.nb. Development and experiments notebook.
6 GST48 source 2022-02-05 17:14:56 2022-02-05 18:02:17  
7 index source 2022-06-24 12:26:51 2022-06-24 12:50:30 This is the index file for the APAI project.
8 Invariance2 source 2022-02-17 11:15:55 2022-02-17 11:15:56 Invariance proof by Roland.
9 Invariance-Scratch source 2022-05-25 09:37:14 2022-06-02 16:39:47  
10 MakeData3-14 source 2022-02-14 09:32:49 2022-02-14 11:15:42 Compute and store ρ, Kh, and HOMFLYPT for knots with up to 14 crossings
11 Make source 2022-01-09 22:23:29 2022-06-24 06:19:39  
12 OtherAlexanders source 2022-01-23 18:14:22 2022-01-25 13:45:12 Other near-Wirtinger matrices to produce the Alexander polynomial.
13 PABI-InBack source 2022-01-07 14:06:43 2022-01-12 07:16:02 Perturbed Alexander-Burau Invariants. Continues Pensieve://2021-12/, but with the perturbation placed ahead of the bulk.
14 Playing-220818 source 2022-08-18 09:21:56 2022-08-18 09:40:19 Playing near $\rho_1$.
15 Rho1-3-100 source 2022-01-06 12:05:01 2022-01-07 09:29:25 Perturbed Alexander-Burau Invariants. DPG code from PerturbedHeisenberg2.nb at pensieve://Classes/21-1350-KnotTheory/. Continued pensieve://Projects/APerturbedAlexanderInvariant/.
16 Rot source 2022-05-24 10:23:57 2022-05-24 10:24:29 Computing rotation numbers. Based on Common.nb at pensieve://Classes/21-1350-KnotTheory/.
17 SimpleInverses source 2022-02-14 15:02:42 2022-02-14 15:02:42 The two-point function and the modified two-point function both have simple inverses.
18 TimingGST48 source 2022-06-16 11:10:35 2022-06-17 05:42:04  

<< Links >>

  1. Studied 2022-05-24 18:09:40: [q-alg-9605023] Xiao-Song Lin, Feng Tian, Zhenghan Wang- Burau representation and random walk on string links.
  2. Studied 2022-06-16 11:32:24: On the complexity of inverting integer and polynomial matrices - SpringerLink.
  3. Studied 2022-06-29 03:27:14: Car Crashes, Group Extensions and s-Cobordisms - Jerusalem Mathematics Colloquium.

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abstract.tex   AlexanderExperiments.m   APAI.aux   APAI.brf   APAI-Expand.m   APAI.m   APAINB.tex   APAI - Other R.m   APAI.tex   defs.tex   Experiments.m   gRules.tex   Invariance2.m   Invariance-R1l.tex   Invariance-R2c.tex   Invariance-R3-Short.tex   Invariance-R3.tex   Invariance-Rest.tex   Invariance-Scratch.m   MakeData3-14.m   makefile   Make.m   new_aux   old_aux   OtherAlexanders.m   PABI-InBack.m   pd   picins.sty   Playing-220818.m   refs.tex   Rot.m   TimingGST48.m  


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abstract.tex   AlexanderExperiments.m   AlexanderExperiments.nb   APAI.aux   APAI.brf   APAI-Expand.m   APAI-Expand.nb   APAI.m   APAI.nb   APAINB.tex   APAI - Other R.m   APAI - Other R.nb   APAI.pdf   APAI-Scratch.pdf   APAI.tex   Car Crashes, Group Extensions and s-Cobordisms - Jerusalem Mathematics Colloquium.url   defs.tex   Experiments.m   Experiments.nb   Geneva-2206 - Shortcut.lnk   gRules.tex   GST48.nb   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   MakeData3-14.m   MakeData3-14.nb   makefile   Make.m   Make.nb   new_aux   Notebook.pdf   old_aux   On the complexity of inverting integer and polynomial matrices - SpringerLink.url   OtherAlexanders.m   OtherAlexanders.nb   PABI-InBack.m   PABI-InBack.nb   pd   picins.sty   Playing-220818.m   Playing-220818.nb   [q-alg-9605023] Xiao-Song Lin, Feng Tian, Zhenghan Wang- Burau representation and random walk on string links.url   refs.tex   Rho1-3-100.nb   Rot.m   Rot.nb   SimpleInverses.nb   SlideShow2.html   SlideShow.html   TimingGST48.m   TimingGST48.nb   Waco-2203 - Shortcut.lnk