Dror Bar-Natan: AcademicPensieve: Projects: APAI:

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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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SlideShow2.html SlideShow.html / APAI@230322.pdf APAI-Dev.pdf APAI.pdf APAI-Scratch.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	2023-01-03 11:17:25	2023-01-04 03:16:05	$A$ - $A^T$ experiments.
2	AlexanderExperiments	source	2022-09-22 14:33:05	2022-09-25 14:14:25	Alexander Experiments: Modifying the A Matrix.
3	APAI-Expand	source	2022-09-13 20:29:10	2022-09-14 03:27:52	Same as APAI.nb, but with Expand replacing Factor.
4	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.
5	APAI	source	2022-01-08 08:25:07	2023-03-21 21:43:39	Mathematica notebook for A Perturbed Alexander Invariant.
6	GGbar	source	2023-01-13 05:23:20	2023-01-13 08:45:52	Failed $G$ - $\bar{G}$ experiments.
7	GGI	source	2023-01-04 02:45:05	2023-01-04 03:31:12	Failed $G$ - $G-I$ experiments.
8	GGT	source	2023-04-20 15:37:06	2023-04-20 15:58:25	Comparing $G$ and $G^T$.
9	GST48	source	2022-02-05 17:14:56	2022-02-05 18:02:17
10	index	source	2022-06-24 12:26:51	2022-06-24 12:50:30	This is the index file for the APAI project.
11	Invariance2	source	2022-02-17 11:15:55	2022-02-17 11:15:56	Invariance proof by Roland.
12	Invariance-Scratch	source	2022-05-25 09:37:14	2022-06-02 16:39:47
13	Lashings-230105	source	2023-03-02 19:21:44	2023-01-05 07:56:03	The Lashings Matrix.
14	Lashings	source	2023-01-04 07:41:08	2023-04-20 14:34:40	The Lashings Matrix.
15	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
16	Make	source	2022-01-09 22:23:29	2022-06-24 06:19:39
17	OtherAlexanders	source	2022-01-23 18:14:22	2022-01-25 13:45:12	Other near-Wirtinger matrices to produce the Alexander polynomial.
18	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.
19	Playing-220818	source	2022-08-18 09:21:56	2022-08-18 09:40:19	Playing near $\rho_1$.
20	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/.
21	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/.
22	TimingGST48	source	2022-06-16 11:10:35	2022-06-17 05:42:04