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Plabic graphs and zonotopal tilings Pavel Galashin MIT galashin@mit.edu FPSAC 2018, Dartmouth College, July 19, 2018 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 = 12 12 14 14 14 23 23 23 34 34 34 1 1 2

  1. Cyclic zonotopes Definition Cyclic vector configuration: C ( n , d ) := ( v 1 , v 2 , . . . , v n ), where v i = (1 , r i , r 2 i , . . . , r d − 1 ) for some 0 < r 1 < r 2 < · · · < r n ∈ R . i Cyclic zonotope: Z ( n , d ) := Z C ( n , d ) . v 2 v 1 v 3 v 6 v 4 v 5 v 4 v 3 v 2 v 1 z = 1 C (4 , 2) = C (6 , 3) = 0 0 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 6 / 33

  2. Zonotopal tilings Definition A zonotopal tiling of Z V is a polyhedral subdivision ∆ of Z V into smaller zonotopes. ∆ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33

  3. Zonotopal tilings Definition A zonotopal tiling of Z V is a polyhedral subdivision ∆ of Z V into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. ∆ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33

  4. Zonotopal tilings Definition A zonotopal tiling of Z V is a polyhedral subdivision ∆ of Z V into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece Z V ′ is a parallelotope if the vectors in V ′ form a basis of R d . ∆ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33

  5. Zonotopal tilings Definition A zonotopal tiling of Z V is a polyhedral subdivision ∆ of Z V into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece Z V ′ is a parallelotope if the vectors in V ′ form a basis of R d . 234 1234 34 24 124 123 4 14 12 ∅ 1 Vert(∆) ⊂ 2 [ n ] ∆ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33

  6. Zonotopal tilings Definition A zonotopal tiling of Z V is a polyhedral subdivision ∆ of Z V into smaller zonotopes. A zonotopal tiling is fine if all pieces are parallelotopes. A piece Z V ′ is a parallelotope if the vectors in V ′ form a basis of R d . 234 1234 v 2 34 24 124 123 v 2 v 2 4 14 12 v 2 ∅ 1 Vert(∆) ⊂ 2 [ n ] ∆ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 7 / 33

  7. Vertices of zonotopal tilings Fact Number of vertices in a fine zonotopal tiling of Z V equals the number Ind( V ) of linearly independent subsets of V . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33

  8. Vertices of zonotopal tilings Fact Number of vertices in a fine zonotopal tiling of Z V equals the number Ind( V ) of linearly independent subsets of V . � n � � n � � n � + · · · + Ind( C ( n , d )) = + . 0 1 d Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33

  9. Vertices of zonotopal tilings Fact Number of vertices in a fine zonotopal tiling of Z V equals the number Ind( V ) of linearly independent subsets of V . � n � � n � � n � + · · · + Ind( C ( n , d )) = + . 0 1 d 234 1234 34 24 124 123 v 4 v 3 v 2 4 14 12 V = ∆ = v 1 ∅ 1 0 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33

  10. Vertices of zonotopal tilings Fact Number of vertices in a fine zonotopal tiling of Z V equals the number Ind( V ) of linearly independent subsets of V . � n � � n � � n � + · · · + Ind( C ( n , d )) = + . 0 1 d 234 1234 34 24 124 123 v 4 v 3 v 2 4 14 12 V = ∆ = v 1 ∅ 1 0 � 4 � � 4 � � 4 � | Vert(∆) | = 11 . Ind( V ) = + + = 11 , 0 1 2 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 8 / 33

  11. Vertices of zonotopal tilings Question Which collections of subsets of [ n ] can appear as Vert(∆) , where ∆ is a fine zonotopal tiling of Z ( n , 2) ? Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33

  12. Vertices of zonotopal tilings Question Which collections of subsets of [ n ] can appear as Vert(∆) , where ∆ is a fine zonotopal tiling of Z ( n , 2) ? Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33

  13. Vertices of zonotopal tilings Question Which collections of subsets of [ n ] can appear as Vert(∆) , where ∆ is a fine zonotopal tiling of Z ( n , 2) ? Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . 1 2 2 3 4 4 5 5 6 7 7 8 9 9 Strongly separated: S \ T T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33

  14. Vertices of zonotopal tilings Question Which collections of subsets of [ n ] can appear as Vert(∆) , where ∆ is a fine zonotopal tiling of Z ( n , 2) ? Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . 1 2 2 3 4 4 5 5 6 7 7 8 9 9 Strongly separated: S \ T T \ S D ⊂ 2 [ n ] is a strongly separated collection if all S , T ∈ D are strongly separated. Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 9 / 33

  15. Purity phenomenon 1 2 2 3 4 4 5 5 6 7 7 8 9 9 Strongly separated: S \ T T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33

  16. Purity phenomenon 1 2 2 3 4 4 5 5 6 7 7 8 9 9 Strongly separated: S \ T T \ S Proposition (Leclerc–Zelevinsky (1998)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 2) , and maximal by inclusion strongly separated collections D ⊂ 2 [ n ] . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33

  17. Purity phenomenon 1 2 2 3 4 4 5 5 6 7 7 8 9 9 Strongly separated: S \ T T \ S Proposition (Leclerc–Zelevinsky (1998)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 2) , and maximal by inclusion strongly separated collections D ⊂ 2 [ n ] . Corollary (Leclerc–Zelevinsky (1998)) Purity phenomenon: every maximal by inclusion strongly separated collection D ⊂ 2 [ n ] is also maximal by size: � n � � n � � n � |D| = + + . 0 1 2 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 10 / 33

  18. 3D zonotopes v 2 v 1 v 3 v 6 v 4 v 5 z = 1 C (6 , 3) Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 11 / 33

  19. 3D zonotopes 123 123 123 12 12 13 13 13 23 23 23 v 2 v 1 v 3 v 6 v 4 v 5 1 1 1 2 2 2 z = 1 3 3 3 ∅ ∅ ∅ Z (3 , 3) C (6 , 3) Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 11 / 33

  20. 3D zonotopes: Z (4 , 3) 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 12 12 14 14 14 23 23 23 34 34 34 1 1 2 2 2 4 4 4 3 3 3 ∅ ∅ ∅ Z (4 , 3) Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 12 / 33

  21. 3D zonotopes: Z (4 , 3) 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 12 12 14 14 14 23 23 23 34 34 34 1 1 2 2 2 4 4 4 3 3 3 ∅ ∅ ∅ Z (4 , 3) Q: How many fine zonotopal tilings? Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 12 / 33

  22. Chord separation Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33

  23. Chord separation Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Strongly separated: 1 2 2 3 4 4 5 5 6 7 7 8 9 9 S \ T T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33

  24. Chord separation Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Definition (G. (2017)) S , T ⊂ [ n ] are chord separated if there is no i < j < k < ℓ such that i , k ∈ S \ T and j , ℓ ∈ T \ S (or vice versa) . Strongly separated: 1 2 2 3 4 4 5 5 6 7 7 8 9 9 S \ T T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33

  25. Chord separation Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Definition (G. (2017)) S , T ⊂ [ n ] are chord separated if there is no i < j < k < ℓ such that i , k ∈ S \ T and j , ℓ ∈ T \ S (or vice versa) . Strongly separated: Chord separated: 3 4 4 S \ T 2 2 5 5 1 2 2 3 4 4 5 5 6 7 7 8 9 9 1 1 6 S \ T T \ S 9 7 7 8 T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33

  26. Chord separation Definition (Leclerc–Zelevinsky (1998)) S , T ⊂ [ n ] are strongly separated if there is no i < j < k such that i , k ∈ S \ T and j ∈ T \ S (or vice versa) . Definition (G. (2017)) S , T ⊂ [ n ] are chord separated if there is no i < j < k < ℓ such that i , k ∈ S \ T and j , ℓ ∈ T \ S (or vice versa) . Strongly separated: Chord separated: 3 4 4 S \ T 2 2 5 5 1 2 2 3 4 4 5 5 6 7 7 8 9 9 1 1 6 S \ T T \ S 9 7 7 8 T \ S When | S | = | T | , both definitions are due to Leclerc–Zelevinsky. Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 13 / 33

  27. Chord separation Strongly separated: Chord separated: 3 4 4 S \ T 2 2 5 5 1 2 2 3 4 4 5 5 6 7 7 8 9 9 1 1 6 S \ T T \ S 9 7 7 8 T \ S Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33

  28. Chord separation Strongly separated: Chord separated: 3 4 4 S \ T 2 2 5 5 1 2 2 3 4 4 5 5 6 7 7 8 9 9 1 1 6 S \ T T \ S 9 7 7 8 T \ S Proposition (Leclerc–Zelevinsky (1998)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 2) , and maximal by inclusion strongly separated collections D ⊂ 2 [ n ] . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33

  29. Chord separation Strongly separated: Chord separated: 3 4 4 S \ T 2 2 5 5 1 2 2 3 4 4 5 5 6 7 7 8 9 9 1 1 6 S \ T T \ S 9 7 7 8 T \ S Proposition (Leclerc–Zelevinsky (1998)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 2) , and maximal by inclusion strongly separated collections D ⊂ 2 [ n ] . Theorem (G. (2017)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 3) , and maximal by inclusion chord separated collections D ⊂ 2 [ n ] . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 14 / 33

  30. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  31. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. The only two subsets of { 1 , 2 , 3 , 4 } that are not chord separated: Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  32. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. The only two subsets of { 1 , 2 , 3 , 4 } that are not chord separated: { 1 , 3 } and { 2 , 4 } . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  33. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. The only two subsets of { 1 , 2 , 3 , 4 } that are not chord separated: { 1 , 3 } and { 2 , 4 } . There are exactly two maximal by inclusion chord separated collections D ⊂ 2 [ n ] . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  34. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. The only two subsets of { 1 , 2 , 3 , 4 } that are not chord separated: { 1 , 3 } and { 2 , 4 } . There are exactly two maximal by inclusion chord separated collections D ⊂ 2 [ n ] . 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 Q: How many fine zonotopal tilings? 12 12 14 14 14 23 23 23 34 34 34 1 1 2 2 2 4 4 4 3 3 3 ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  35. Example for n = 4 Chord separation : no i < j < k < ℓ such that i , k ∈ S \ T , j , ℓ ∈ T \ S or vice versa. The only two subsets of { 1 , 2 , 3 , 4 } that are not chord separated: { 1 , 3 } and { 2 , 4 } . There are exactly two maximal by inclusion chord separated collections D ⊂ 2 [ n ] . 1234 1234 1234 124 124 123 123 123 134 134 134 234 234 234 Q: How many fine zonotopal tilings? 12 12 A: Two. 14 14 14 23 23 23 34 34 34 1 1 2 2 2 4 4 4 3 3 3 ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 15 / 33

  36. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 14 14 14 14 14 14 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  37. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 14 14 14 14 14 14 13 13 13 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  38. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 14 14 14 14 14 14 13 13 13 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  39. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 14 14 14 14 14 14 13 13 13 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  40. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 14 14 14 14 14 14 13 13 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  41. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 12 14 14 14 14 14 14 13 13 24 24 24 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  42. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 12 14 14 14 14 14 14 13 13 24 24 24 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  43. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 12 14 14 14 14 14 14 13 13 24 24 24 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  44. Fine zonotopal tilings of Z (4 , 3) 1234 1234 1234 1234 1234 1234 124 124 124 124 124 123 123 123 123 123 123 134 134 134 134 134 134 234 234 234 234 234 234 12 12 12 12 12 14 14 14 14 14 14 13 13 24 24 23 23 23 23 23 23 34 34 34 34 34 34 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 3 3 3 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 16 / 33

  45. Part 2: Plabic graphs

  46. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  47. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  48. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  49. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  50. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  51. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  52. Plabic graphs and strands Definition (Postnikov (2007)) A plabic graph is a planar graph embedded in a disk, with n boundary vertices of degree 1, and the remaining vertices all trivalent and colored black and white. A strand in a plabic graph is a path that turns right at each black vertex turns left at each white vertex Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 18 / 33

  53. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  54. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: No closed strands Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  55. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: No closed No strand strands intersects itself Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  56. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: No closed No strand No “bad double strands intersects itself crossings” Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  57. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: No closed No strand No “bad double “Good double strands intersects itself crossings” crossings” are OK! Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  58. ( k , n )-plabic graphs Definition (Postnikov (2007)) A plabic graph is reduced if it contains: No closed No strand No “bad double “Good double strands intersects itself crossings” crossings” are OK! A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 19 / 33

  59. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  60. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  61. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  62. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  63. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  64. ( k , n )-plabic graphs Definition (Postnikov (2007)) A ( k , n ) -plabic graph is a reduced plabic graph such that: the strand that starts at i ends at i + k modulo n for all i . 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 20 / 33

  65. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. 5 1 4 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  66. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  67. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  68. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  69. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  70. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  71. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  72. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 5 1 2 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  73. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 5 1 1 5 1 3 2 3 4 2 3 5 4 5 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  74. Face labels Postnikov (2007): each ( k , n )-plabic graph has k ( n − k ) + 1 faces. Scott (2005): label each face of a ( k , n )-plabic graph by a k -element set: include j in this set iff the face is to the left of the strand i → j . 1 2 1 2 5 1 1 5 1 5 1 3 1 3 2 3 2 3 4 2 3 5 3 5 4 5 4 5 3 4 3 4 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 21 / 33

  75. Plabic graphs and chord separation Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33

  76. Plabic graphs and chord separation Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Proved independently by Danilov–Karzanov–Koshevoy (2010) and Oh–Postnikov–Speyer (2011). Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33

  77. Plabic graphs and chord separation Conjecture (Leclecrc–Zelevinsky (1998), Scott (2005)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Proved independently by Danilov–Karzanov–Koshevoy (2010) and Oh–Postnikov–Speyer (2011). Theorem (Oh–Postnikov–Speyer (2011)) The map G �→ Faces( G ) is a bijection ∗ between: ( k , n ) -plabic graphs, and � [ n ] � maximal by inclusion chord separated collections D ⊂ . k Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 22 / 33

  78. Contradiction? Corollary (Oh–Postnikov–Speyer (2011)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33

  79. Contradiction? Corollary (Oh–Postnikov–Speyer (2011)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Theorem (G. (2017)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 3) , and maximal by inclusion chord separated collections D ⊂ 2 [ n ] . Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33

  80. Contradiction? Corollary (Oh–Postnikov–Speyer (2011)) � [ n ] � Every maximal by inclusion chord separated collection D ⊂ has size k k ( n − k ) + 1 . Theorem (G. (2017)) The map ∆ �→ Vert(∆) is a bijection between: fine zonotopal tilings ∆ of Z ( n , 3) , and maximal by inclusion chord separated collections D ⊂ 2 [ n ] . Corollary (G. (2017)) Every maximal by inclusion chord separated collection D ⊂ 2 [ n ] has size � n � � n � � n � � n � Ind( C ( n , 3)) = + + + . 0 1 2 3 Pavel Galashin (MIT) Plabic graphs and zonotopal tilings FPSAC 2018, 07/19/2018 23 / 33

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