Duality for Algebras of the Connected Planar Wiring Diagrams Operad Owen Biesel Carleton College obiesel@carleton.edu May 23, 2019 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Combining Resistance R 1 R 2 R 1 + R 2 ∼ = R 1 R 1 R 2 R 1 + R 2 ∼ = R 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Combining Resistance R 1 R 2 R 1 + R 2 ∼ = R 1 � 1 � − 1 + 1 R 1 R 2 ∼ = R 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Conductance is Inverse Resistance G = 1 / R R = 2 Ω G = 0 . 5 ℧ � resistance conductance Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Combining Conductance � 1 � − 1 + 1 G 1 G 2 G 1 G 2 ∼ = G 1 G 1 + G 2 ∼ = G 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Combining Maximum Flow Rates F 2 F 1 min( F 1 , F 2 ) ∼ = F 1 ∼ = F 1 + F 2 F 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Combining Minimum Path Lengths D 1 D 2 D 1 + D 2 ∼ = min( D 1 , D 2 ) D 1 ∼ = D 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Series and parallel formulas Series Parallel ( R − 1 + R − 1 2 ) − 1 Resistance R 1 + R 2 1 ( G − 1 + G − 1 2 ) − 1 Conductance G 1 + G 2 1 Max Flow Rate min( F 1 , F 2 ) F 1 + F 2 Min Path Length D 1 + D 2 min( D 1 , D 2 ) Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Series and parallel connections are dual Not quite the usual “dual graph.” Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Connected circular planar graphs Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Placing the dual vertices Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Placing the dual edges Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
The dual graph Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Dual Connected Circular Planar Graphs Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Resistance and conductance are “dual” Resistances Conductances 4 ℧ 1 ℧ 1 Ω 4 Ω 4 Ω 1 ℧ 1 Ω 4 ℧ 3 Ω 1 Ω 3 ℧ 1 ℧ Effective resistance = 157 Effective conductance = 157 29 Ω 29 ℧ Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Max flow rate and min path length are “dual” Edge lengths Max flow rates 4 1 1 4 1 4 4 1 3 1 3 1 Min path length = 3 Overall max flow rate = 3 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Main theorem Theorem (B.—, 2019) (Connected circular planar) graphs form an algebra of the operad P lan of “connected planar wiring diagrams.” “Max flow rate,” “min path length,” “effective resistance,” and “effective conductance” are all morphisms between P lan-algebras. P lan has a duality automorphism giving every algebra a “dual” algebra. The algebra of graphs is isomorphic to its dual algebra, and the isomorphism sends a graph to its dual graph. The dual of the “max flow rate” morphism is “min path length,” and the dual of “effective resistance” is “effective conductance.” Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing together circular planar graphs ? � Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing with a planar wiring diagram = Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
A connected planar wiring diagram This is a morphism in the operad P lan of connected planar wiring diagrams . Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Series and parallel wiring diagrams Series Parallel Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
The operad of connected planar wiring diagrams Definition (B.—, 2019) The (symmetric, coloured) operad P lan : objects are circularly ordered finite sets ( X , θ ). θ ( x ) x X Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
The operad of connected planar wiring diagrams Definition (continued) morphisms from ( X 1 , θ 1 ) , . . . , ( X n , θ n ) to ( Y , ϕ ) are planar wiring diagrams with the X i on the inside and Y on the outside: Every “cable” has ≤ 1 X 1 X 2 element of Y . Every “face” has ≤ 1 X 3 arc of outer circle. Y Lemma: This really does define an operad! Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Relationships to other operads P lan has: A forgetful map to Spivak’s operad of all wiring diagrams. P lan inherits several algebras from there, like flows and potentials. However, Spivak’s operad does not have a duality automorphism. An inclusion map to Jones’s “planar algebras” operad. P lan inherits its duality automorphism Jones’s “1-click” automorphism, but Jones’s operad has too many morphisms for circular planar graphs to be an algebra. Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
P lan -algebras A P lan -algebra A assigns: to each circularly ordered set ( X , θ ) a set A ( X , θ ), and to each morphism ( X 1 , θ 1 ) , . . . , ( X n , θ n ) → ( Y , ϕ ) a function n � A ( X i , θ i ) → A ( Y , ϕ ) . i =1 These describe what may be inserted into the slots of a wiring diagram and how they glue together. A P lan − → Op (Set) Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Example P lan algebras: G and G (0 , ∞ ) G ( X , θ ) = set of connected circular planar graphs with boundary vertices ( X , θ ). G (0 , ∞ ) ( X , θ ): same, but with edges weighted by positive real numbers. Lemma: G is generated by the single element satisfying a single relation. That makes it easy to describe algebra morphisms out of G ! Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Example P lan algebra: Π Π( X , θ ) = set of planar (noncrossing) partitions of ( X , θ ). = Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Example P lan -algebra: potential sets A potential on ( X , θ ) is a function X → R up to overall additive constant: 1 101 0 100 X 2 = X 102 − 2 98 4 104 V ( X , θ ) is the set of potentials on X . Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing compatible potentials 1 4 − 1 0 1 3 2 2 = ? 0 2 1 1 4 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing compatible potentials 1 3 − 1 0 0 2 1 3 2 1 2 = − 1 0 2 1 1 1 1 4 4 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing potential sets Not all potentials can be glued, so V is only a relational P lan -algebra. But P ( V ) : ( X , θ ) �→ the set of subsets of V ( X , θ ) is an actual P lan -algebra! Send a collection of potential sets to the set of potentials obtained by gluing compatible members. Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Min path length: G (0 , ∞ ) → P ( V ) Weights on a graph � distances � potential set: D � { ( a , b ) ∈ R 2 | | a − b | ≤ D } Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Min path length: G (0 , ∞ ) → P ( V ) D 2 D 1 � { ( a , b ) ∈ R 2 | ∃ x ∈ R such that D 3 | a − x | ≤ D 1 , | x − b | ≤ D 2 , | x − b | ≤ D 3 } = { ( a , b ) ∈ R 2 | | a − b | ≤ D 1 + min( D 2 , D 3 ) } “Min path length” is an algebra morphism G (0 , ∞ ) → P ( V )! Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Example P lan -algebras: Flow Sets The algebra P ( F ) of flow sets : A flow on ( X , θ ) is a sum-zero function X → R : 0 . 5 − 1 X 3 − 2 − 0 . 5 F ( X , θ ) =the set of flows on X . Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing compatible flows − 1 0 − 1 3 − 2 1 − 1 0 − 1 1 1 = − 1 − 1 1 1 2 1 1 − 2 − 2 Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Gluing flow sets Not all flows can be glued, so F is only a relational P lan -algebra. But P ( F ) : ( X , θ ) �→ the set of subsets of F ( X , θ ) is an actual P lan -algebra! Send a collection of flow sets to the set of flows obtained by gluing compatible members. Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Max flow rate: G (0 , ∞ ) → P ( F ) Weights on a graph � possible flows: F � { ( a , − a ) ∈ R 2 | | a | ≤ F } Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
Max flow rate: G (0 , ∞ ) → P ( F ) F 2 F 1 � { ( a , b ) ∈ R 2 | ∃ x , y , z ∈ R such that F 3 | x | ≤ F 1 , | y | ≤ F 2 , | z | ≤ F 3 , x = a , y + z = b , x + y + z = 0 } = { ( a , − a ) ∈ R 2 | | a | ≤ min( F 1 , F 2 + F 3 ) } “Max flow rate” is a P lan -algebra morphism G (0 , ∞ ) → P ( F )! Owen Biesel (Carleton) Duality for Planar Wiring Diagrams May 23, 2019
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