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Abstract. Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful "gliding ideas" in the literature - sometimes in disguise - such as the Drinfel'd double construction, Enriquez's work on quantization of Lie bialgebras, and Audoux and Meilhan's classification of welded homotopy links.
Archive Data figs nb2tex_pdfs OU-200716-Dror OU-201105-Zsuzsi OU-210101-Roland OU-210118-Dror OU-221110-JKTR / Talks: Bangalore-2003 Projects: DPG Talks: MoscowByWeb-2004 Talks: TrendsInLDT-2005 BorrBraidOUDiagram2.pdf CinnamonRoll.pdf CJM_Review_and_Response.pdf main.pdf NYJM_Reviews.pdf OU.pdf OUTagged.pdf report_M_on_v1.pdf ReverseGamma.pdf Waterfall.pdf / JKTRRefereeReport.txt recycling.txt<< Mathematica Notebooks >>
| Notebook (.pdf) | Source (.nb) | Created | Last Modified | Summary | |
|---|---|---|---|---|---|
| 1 | BraidInBurauKernel | source | 2024-03-17 10:28:59 | 2023-05-26 11:58:33 | |
| 2 | BraidsForWiest | source | 2024-03-17 10:28:59 | 2020-05-28 05:57:21 | Extraction graphs for braids picked by Bert Wiest. |
| 3 | ChterentalsBraid | source | 2024-03-17 10:28:59 | 2020-05-29 15:01:19 | Studying Chterental's ker(\[Psi]) virtual braid. |
| 4 | CountingClassicalBraids | source | 2024-03-17 10:28:59 | 2023-05-26 12:01:31 | Counting classical braids by computing $\bar\Gamma$ of all classical proud braids. |
| 5 | DiamondGraphs | source | 2024-03-17 10:28:59 | 2020-05-31 08:51:28 | OU matters around the Diamond Lemma. |
| 6 | DiamondMining | source | 2024-03-17 10:28:59 | 2020-05-13 17:15:35 | An attempt to "mine" all the diamonds within an Extraction Graph. Fails because the notion of a diamond is not entirely well-defined, as in ExtranctionGraph[BR[3,{-1,-2,-2,-2,-1}]]. |
| 7 | Diamond | source | 2024-03-17 10:28:59 | 2020-05-31 08:53:46 | OU matters around the Diamond Lemma. |
| 8 | EquivalenceRelations | source | 2024-03-17 10:28:59 | 2020-03-21 14:06:04 | A package to manage equivalence relations. |
| 9 | ExtractionGraphsForAllBraids | source | 2024-03-17 10:28:59 | 2020-07-06 11:53:01 | All extraction graphs of all small-complexity classical braids. |
| 10 | ExtractionGraphsForKnotBraids | source | 2024-03-17 10:28:59 | 2020-05-03 07:30:56 | Extraction graphs for braids representing the Rolfsen table knots. |
| 11 | ExtractionGraphsGallery | source | 2024-03-17 10:28:59 | 2020-05-16 06:04:42 | A gallery of fun or noteworthy extraction graphs. |
| 12 | GraphLayouts | source | 2024-03-17 10:28:59 | 2020-07-06 11:55:37 | Experiments with graph layouts. |
| 13 | index | source | 2024-03-17 10:28:59 | 2022-11-10 12:46:56 | This is the index file for the 2020-03/OU project; it also produces WordCloud.png. |
| 14 | nb2tex | source | 2024-03-17 10:28:59 | 2021-02-04 18:07:25 | nb2tex for the OU project. |
| 15 | OU-Programs | source | 2024-03-17 10:28:59 | 2020-06-03 09:11:04 | The OU Programs (minus counting by gens and rels). |
| 16 | ReverseGamma | source | 2024-03-17 10:28:59 | 2020-07-11 11:59:10 | Generating Figure fig:ReverseGamma. |
| 17 | SomeComputations | source | 2024-03-17 10:28:59 | 2021-02-04 18:06:27 | The primary program accompanying "Over then Under Tangles", by Dror Bar-Natan, Zsuzsanna Dancso, and Roland van der Veen. |
| 18 | TabulatingClassicalBraids | source | 2024-03-17 10:28:59 | 2020-07-06 09:00:25 | Tabulating classical braids. |
| 19 | TabulatingVirtualBraids | source | 2024-03-17 10:28:59 | 2020-06-29 07:50:51 | Tabulating virtual braids. |
| 20 | TheWorstBraids | source | 2024-03-17 10:28:59 | 2020-07-06 19:03:11 | Finding the braids for which the complexity of $\bar{\Gamma}$ grows in the fastest way. |
| 21 | VPBByGensAndRels | source | 2024-03-17 10:28:59 | 2020-07-06 12:24:12 | Counting virtual pure braids using generators and relations. Slow and unreliable, produces only upper bounds, yet agrees with the OU computations. |
<< Links >>
- Studied 2020-04-21 17:19:02: Diamond Lemma - MathOverflow.
- Studied 2020-05-11 15:34:54: Cinnamon Roll Icons - Download Free Vector Icons - Noun Project.
- Studied 2020-07-16 16:32:28: Dror Bar-Natan- Publications.
- Studied 2020-07-21 05:50:26: [2007.09828] Over then Under Tangles.
- Studied 2021-04-20 07:55:18: [1506.05283] Paolo Bellingeri, Bruno Aaron Cisneros de La Cruz, Luis Paris- A simple solution to the word problem for virtual braid groups.
<< Images >>
