Flip cycles in plabic graphs
Abstract Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $$\text {Gr}^{\ge 0}(n,k)$$Gr≥0(n,k). Any two plabic graphs for the same positroid cell can be related by a sequence o...
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/131399 |
| _version_ | 1826211884961890304 |
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| author | Balitskiy, Alexey Wellman, Julian |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Balitskiy, Alexey Wellman, Julian |
| author_sort | Balitskiy, Alexey |
| collection | MIT |
| description | Abstract
Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $$\text {Gr}^{\ge 0}(n,k)$$Gr≥0(n,k). Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope Z(n, 3). Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem. |
| first_indexed | 2024-09-23T15:12:58Z |
| format | Article |
| id | mit-1721.1/131399 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:12:58Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1313992023-10-06T20:14:10Z Flip cycles in plabic graphs Balitskiy, Alexey Wellman, Julian Massachusetts Institute of Technology. Department of Mathematics Abstract Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $$\text {Gr}^{\ge 0}(n,k)$$Gr≥0(n,k). Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope Z(n, 3). Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem. 2021-09-20T17:16:55Z 2021-09-20T17:16:55Z 2020-02-17 2020-09-24T21:11:36Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131399 Selecta Mathematica. 2020 Feb 17;26(1):15 en https://doi.org/10.1007/s00029-020-0544-1 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Balitskiy, Alexey Wellman, Julian Flip cycles in plabic graphs |
| title | Flip cycles in plabic graphs |
| title_full | Flip cycles in plabic graphs |
| title_fullStr | Flip cycles in plabic graphs |
| title_full_unstemmed | Flip cycles in plabic graphs |
| title_short | Flip cycles in plabic graphs |
| title_sort | flip cycles in plabic graphs |
| url | https://hdl.handle.net/1721.1/131399 |
| work_keys_str_mv | AT balitskiyalexey flipcyclesinplabicgraphs AT wellmanjulian flipcyclesinplabicgraphs |