Dror Bar-Natan: AcademicPensieve: Projects: APAI:

(ↄ) | Dror Bar-Natan: AcademicPensieve: Projects:

About Recent Changes This Month Random

<< APAI >>

Jump to: Mathematica Notebooks | Links | Images | Other Files

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 suggests that they should have a topological explanation. This we do not have yet.

<< Subfolders / Shortcuts >>

Archive Data figs Snips / Talks: Geneva-2206 Projects: HigherRank Talks: Oaxaca-2210 People: VanDerVeen Talks: Waco-2203

<< HTML / PDF / TXT >>

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	Sun 17 Mar 2024 10:28:44	Wed 4 Jan 2023 02:16:05	$A$ - $A^T$ experiments.
2	AlexanderExperiments	source	Sun 17 Mar 2024 10:28:44	Sun 25 Sep 2022 13:14:25	Alexander Experiments: Modifying the A Matrix.
3	APAI-Expand	source	Sun 17 Mar 2024 10:28:44	Wed 14 Sep 2022 02:27:52	Same as APAI.nb, but with Expand replacing Factor.
4	APAI-InvarianceForLinks	source	Tue 9 Apr 2024 08:43:56	Tue 9 Apr 2024 08:47:54	Mathematica notebook for A Perturbed Alexander Invariant.
5	APAILinks2	source	Mon 1 Apr 2024 15:49:53	Mon 1 Apr 2024 15:58:41	APAI for links, using the adjugate matrix (by Roland).
6	APAILinks	source	Sun 17 Mar 2024 10:28:44	Thu 9 Nov 2023 06:34:23	A version of $\rho_1$ for links with some testing, by Roland.
7	APAI - Other R	source	Sun 17 Mar 2024 10:28:44	Mon 19 Sep 2022 14:50:56	Mathematica notebook for A Perturbed Alexander Invariant, with revised Rs.
8	APAI	source	Sun 17 Mar 2024 10:28:44	Sun 14 Apr 2024 13:02:12	Mathematica notebook for A Perturbed Alexander Invariant.
9	APAI-Positivity (1)	source	Sun 17 Mar 2024 10:28:44	Tue 5 Mar 2024 17:00:43
10	APAI-Positivity	source	Sun 17 Mar 2024 10:28:44	Tue 5 Mar 2024 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	Sun 17 Mar 2024 10:28:44	Fri 4 Aug 2023 01:37:16	Just colour-coded $A$-matrices.
12	GGbar	source	Sun 17 Mar 2024 10:28:44	Fri 13 Jan 2023 07:45:52	Failed $G$ - $\bar{G}$ experiments.
13	GGI	source	Sun 17 Mar 2024 10:28:44	Wed 4 Jan 2023 02:31:12	Failed $G$ - $G-I$ experiments.
14	GGT	source	Sun 17 Mar 2024 10:28:44	Thu 20 Apr 2023 14:58:25	Comparing $G$ and $G^T$.
15	GST48	source	Sun 17 Mar 2024 10:28:44	Sat 5 Feb 2022 17:02:17
16	index	source	Sun 17 Mar 2024 10:28:44	Fri 24 Jun 2022 11:50:30	This is the index file for the APAI project.
17	Invariance2	source	Sun 17 Mar 2024 10:28:44	Thu 17 Feb 2022 10:15:56	Invariance proof by Roland.
18	Invariance-Scratch	source	Sun 17 Mar 2024 10:28:44	Thu 2 Jun 2022 15:39:47
19	Lashings-230105	source	Sun 17 Mar 2024 10:28:44	Thu 5 Jan 2023 06:56:03	The Lashings Matrix.
20	Lashings	source	Sun 17 Mar 2024 10:28:44	Thu 20 Apr 2023 13:34:40	The Lashings Matrix.
21	MakeData3-14	source	Sun 17 Mar 2024 10:28:44	Mon 14 Feb 2022 10:15:42	Compute and store ρ, Kh, and HOMFLYPT for knots with up to 14 crossings
22	Make	source	Sun 17 Mar 2024 10:28:44	Sun 14 Apr 2024 13:02:42
23	OtherAlexanders	source	Sun 17 Mar 2024 10:28:44	Tue 25 Jan 2022 12:45:12	Other near-Wirtinger matrices to produce the Alexander polynomial.
24	PABI-InBack	source	Sun 17 Mar 2024 10:28:44	Wed 12 Jan 2022 06:16:02	Perturbed Alexander-Burau Invariants. Continues Pensieve://2021-12/, but with the perturbation placed ahead of the bulk.
25	Palindromicity-GaussianLanguage	source	Sun 17 Mar 2024 10:28:44	Fri 22 Dec 2023 06:27:44	Palindromicity by flipping and manipulating, in Gaussian integration language.
26	Palindromicity-MatrixLanguage	source	Sun 17 Mar 2024 10:28:44	Wed 20 Dec 2023 04:29:57	Palindromicity by flipping and manipulating, in matrix language.
27	PerturbedGaussianIntegration-Invariance	source	Sun 17 Mar 2024 10:28:44	Wed 28 Feb 2024 12:20:05	Invariance of $\rho_1$ in the language of perturbed Gaussian Integration.
28	PerturbedGaussianIntegration	source	Sun 17 Mar 2024 10:28:44	Wed 8 May 2024 02:07:44	Computing and playing with $\rho_1$ in the language of perturbed Gaussian Integration.
29	Playing-220818	source	Sun 17 Mar 2024 10:28:44	Thu 18 Aug 2022 08:40:19	Playing near $\rho_1$.
30	Playing-231222	source	Sun 17 Mar 2024 10:28:44	Fri 22 Dec 2023 09:14:29	The top row of $G$ isn't friendly.
31	Rho1-3-100	source	Sun 17 Mar 2024 10:28:44	Fri 7 Jan 2022 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	Tue 26 Mar 2024 12:51:50	Tue 26 Mar 2024 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	Sun 17 Mar 2024 10:28:44	Tue 24 May 2022 09:24:29	Computing rotation numbers. Based on Common.nb at pensieve://Classes/21-1350-KnotTheory/.
34	TimingGST48	source	Sun 17 Mar 2024 10:28:44	Fri 17 Jun 2022 04:42:04