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Click to download the main Mathematica packages used within: FreeLie.m and AwCalculus.m.
Abstract. In the previous three papers in this series, [WKO1]-[WKO3], Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what remains is a study of a certain number of equations written in a family of spaces ${\mathcal A}^w$, closely related to degree-completed free Lie algebras and to degree-completed spaces of cyclic words.
The purpose of this paper is to introduce mathematical and computational tools that enable explicit computations (up to a certain degree) in these ${\mathcal A}^w$ spaces and to use these tools to solve the said equations and verify some properties of their solutions, and as a consequence, to carry out the computation (up to a certain degree) of certain knot-theoretic invariants discussed in [WKO1]-[WKO3] and in my related paper [KBH].
Front page links: WKO4.pdf, FreeLie.m, AwCalculus.m, WKO4Session (nb, pdf), bch (nb, pdf), 817 (nb, pdf), Borromean (nb, pdf), VCapSolution (nb, pdf), KVDirect (nb, pdf), dims (nb, pdf), dims1 (nb, pdf), Phi (nb, pdf).
Archive ComputerTalk figs / 2013-04 2014-01 2013-11: DoubleTree CheatSheetFreeLie.pdf WKO4.pdf WKO4Tagged.pdf / abstract.txt listing.txt recycling.txt
<< Notebook Pages >>
| Page | Created (UT) | Last Modified (UT) | |
|---|---|---|---|
| 1 | Goals | 2014-02-28 22:04:37 | 2014-02-28 22:10:27 |
| 2 | $\Gamma$, $\Lambda$ | 2014-03-13 15:41:49 | 2014-03-19 18:38:12 |
| 3 | Proof of the Conversion Theorem | 2014-06-25 18:57:23 | 2015-10-05 23:42:04 |
| 4 | --- Comparing Products | 2014-08-06 00:53:54 | 2014-08-06 14:03:26 |
| 5 | --- In Pictures | 2014-08-07 11:05:36 | 2014-08-07 11:24:30 |
| 6 | --- Why Same? | 2014-09-02 13:25:47 | 2014-09-02 13:29:45 |
<< Mathematica Notebooks >>
| Notebook (.pdf) | Source (.nb) | Created | Last Modified | Summary | |
|---|---|---|---|---|---|
| 1 | 817 | source | 2024-03-17 11:29:26 | 2015-12-20 11:45:34 | Fuller output of for $8_{17}$. |
| 2 | AwCalculus | source | 2024-03-17 11:29:26 | 2015-09-10 19:21:15 | Aw-calculus programs for the WKO4 paper. |
| 3 | AwCalculus-Testing | source | 2024-03-17 11:29:26 | 2015-02-12 12:50:58 | Testing the AwCalculus package. |
| 4 | bch | source | 2024-03-17 11:29:26 | 2015-12-21 14:54:52 | Fuller output of bch. Even fuller: bch-to-20.m, bch-to-22.m, bch-to-23.m. |
| 5 | bch-stats | source | 2024-03-17 11:29:26 | 2016-01-11 13:54:07 | Fuller output of bch. Even fuller: bch-to-20.m, bch-to-22.m, bch-to-23.m. |
| 6 | Borromean | source | 2024-03-17 11:29:26 | 2015-12-21 19:20:02 | Fuller output of for the Borromean tangle. Even fuller: Borromean-to-7.m |
| 7 | CellExport | source | 2024-03-17 11:29:26 | 2017-07-06 10:10:02 | A program for exporting tagged cells, continues pensieve://2013-03/, continued pensieve://Projects/PPSA/. |
| 8 | CheatSheetFreeLie-Verification | source | 2024-03-17 11:29:26 | 2014-09-09 15:14:12 | Cheat Sheet Free Lie Verification; continues CheatSheetJ-Verification @ pensieve://2014-01/. |
| 9 | dims1 | source | 2024-03-17 11:29:26 | 2015-05-31 04:17:10 | The linearized KV and twist equations. |
| 10 | dims2 | source | 2024-03-17 11:29:26 | 2015-05-29 16:20:45 | The linearized tree-level KV and twist equations. |
| 11 | dims3 | source | 2024-03-17 11:29:26 | 2015-05-29 16:19:56 | The linearized tree-level KV equations. |
| 12 | dims | source | 2024-03-17 11:29:26 | 2015-05-29 06:38:42 | The linearized KV equations. |
| 13 | FlipSymmetry | source | 2024-03-17 11:29:26 | 2015-04-27 08:35:18 | Testing if vertical flip is a symmetry of SolKV. |
| 14 | FollowingAlbertHarinckTorossian | source | 2024-03-17 11:29:26 | 2015-05-28 11:47:04 | Calculations following Albert-Harinck-Torossian's arXiv:0802.2049. |
| 15 | FreeLie | source | 2024-03-17 11:29:26 | 2018-07-20 12:38:56 | A free-Lie calculator, pensieve://Projects/WKO4/ branch (most current), continues pensieve://2014-01/. Now understands the Drinfel'd-Kohno algebra. |
| 16 | FreeLie-Testing | source | 2024-03-17 11:29:26 | 2018-07-20 13:19:49 | Testing the FreeLie package. |
| 17 | index | source | 2024-03-17 11:29:26 | 2016-02-05 10:36:10 | |
| 18 | KVDirect | source | 2024-03-17 11:29:26 | 2015-08-07 16:35:24 | Fuller output of for the direct KV solution. Even fuller: KVDirect-to-13.m. |
| 19 | Make | source | 2024-03-17 11:29:26 | 2016-07-25 19:57:47 | |
| 20 | muCalculus | source | 2024-03-17 11:29:26 | 2014-02-24 15:45:36 | μ-calculus programs, continues pensieve://2013-11/DoubleTree/. |
| 21 | PhiIn2Poles | source | 2024-03-17 11:29:26 | 2023-04-06 01:45:52 | Φ equations in the 2-poles 2-strands universe. |
| 22 | Phi | source | 2024-03-17 11:29:26 | 2015-04-17 07:43:14 | Fuller output of for a Drinfel'd associator Φ. |
| 23 | Recycling | source | 2024-03-17 11:29:26 | 2015-02-05 19:33:20 | |
| 24 | SolvingForTheUnitarityOfPhiV@230706 | source | 2024-03-17 11:29:26 | 2023-07-05 21:53:26 | Testing if KV is equivalent to R4 + Unitarity of $\Phi_V$. |
| 25 | SolvingForTheUnitarityOfPhiV | source | 2024-03-17 11:29:26 | 2023-07-03 21:50:02 | Testing if KV is equivalent to R4 + Unitarity of $\Phi_V$. |
| 26 | Trialities | source | 2024-03-17 11:29:26 | 2015-08-05 20:55:00 | Figuring out triality(s). |
| 27 | UnitarityOfPhiV | source | 2024-03-17 11:29:26 | 2023-07-05 21:30:56 | Testing the equivalence KV <-> (R4 and ||Phi_V||)==1. |
| 28 | V-Alone_Solution | source | 2024-03-17 11:29:26 | 2023-09-26 15:27:56 | Solving for V alone (with Kuno). Continued pensieve://People/Kuno. |
| 29 | VCapFlipSolution | source | 2024-03-17 11:29:26 | 2015-05-04 09:09:24 | Solving R4, Unitarity, Cap, and Vertical Flip. Data file: R4UniCapVFlipSolution-to-12.m. |
| 30 | VCapSolution | source | 2024-03-17 11:29:26 | 2023-07-03 21:26:26 | Fuller output of for V and Cap. Even fuller: VCapSolution-to-12.m. |
| 31 | VCapSolutionSatisfyNewQ | source | 2024-03-17 11:29:26 | 2023-10-08 12:08:39 | Do solutions of R4, unitarity and cap satisfy the new $\bar{V}^{21}=S_1S_2V^{-1}$?. |
| 32 | VCapSolutionWithNonTrivialWens | source | 2024-03-17 11:29:26 | 2023-10-16 11:24:42 | Solving R4, Cap, and Unitarity without assuming $W=1$. With Dancso, October 2023. |
| 33 | VCapTwistSolution | source | 2024-03-17 11:29:26 | 2015-04-19 10:39:14 | Solving R4, Unitarity, Cap, and Twist. |
| 34 | WKO4Session | source | 2024-03-17 11:29:26 | 2016-07-25 19:55:18 | Calculations appearing in the WKO4 paper. |
<< Links >>
- Studied 2014-03-26 13:50:02: CTAN Package bigfoot.
- Studied 2015-02-11 10:16:06: Int. J. of Alg. & Computation.
- Studied 2015-11-18 22:33:45: [1511.05624] Finite Type Invariants of w-Knotted Objects IV Some Computations.
- Studied 2017-05-04 16:15:18: Journal of Applied and Computational Topology.
<< Images >>
