Flip cycles in plabic graphs
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29_2020_544_ReferencePDF.pdf
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1.63 MB
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Adobe PDF
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628aab3e1aea1d6be83d97e85fd9ad81
Author(s)
Balitskiy, Alexey
Wellman, Julian
Date Issued
February 17, 2020
Publisher
Springer International Publishing
Citation
Selecta Mathematica. 2020 Feb 17;26(1):15
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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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.
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DOI of Published Version
https://doi.org/10.1007/s00029-020-0544-1
