The loop graph, evaluated

The accompanying paper's worked example of Definition 2.1 (§2.4), carried out in the flag model. With k = 2 and ℓ = 1, the functional h(θ) of charPolyFunctional has p_{h(θ)} equal to the characteristic polynomial G ↦ det(θ I − A_G) on graphs without free circles (Regts–Sevenster, arXiv:1807.04494, Proposition 10), and the graph L with one vertex and one loop has A_L = (2). So p_{h(θ)}(L) must be θ − 2, and mixedPartition_loopGraph is that evaluation.

The example is a convention check. Reaching θ − 2 exercises, in one number, the two incidences of a loop at its vertex, the Eulerian condition, the circuit sign, the distinction between a loop and a free circle, and the η-convention that makes the two odd colourings contribute through a common basis vector. A sign error in any one of them changes the answer.

The graph

def RS.loopGraph : ClosedFragment

The loop graph L: one vertex, one edge, both of whose flags are attached to that vertex. No boundary labels, and no free circles — the loop is an edge, not a circle.

Equations

• One or more equations did not get rendered due to their size.

The functional

def RS.charPolyFunctional (θ : ℂ) : MixedFunctional 2 1

The paper's example functional h(θ), with k = 2 and ℓ = 1. On the paper's basis it is h(e₁^⊙i) = θ, h(e₁^⊙i ⊙ e₂) = √-1, h(e₁^⊙i ⊗ ξ₁ ∧ η₁) = 1, and zero elsewhere.

The odd value carried here is the coefficient on the sorted wedge ξ₁ ∧ ξ₂, which is −1: the symplectic partner of ξ₁ is η₁ = −ξ₂, so ξ₁ ∧ η₁ = −ξ₁ ∧ ξ₂. These are the partner conventions of the paper's §2.4, and the two odd colourings each contribute −1 below.

Equations

• RS.charPolyFunctional θ μ w = if Multiset.count 1 μ = 0 then if w = ∅ then θ else if w = Finset.univ then -1 else 0 else if Multiset.count 1 μ = 1 ∧ w = ∅ then Complex.I else 0

The two Eulerian subsets

The loop has two half-edges at its vertex, so both the empty subset and the whole loop are Eulerian, and no other flag set is closed under the edge pairing.

Circuit counts

The colourings

Small facts about the graph

The Eulerian condition

The value of the empty subset

Nothing participates, so there is no odd sector and no circuit sign. The loop's two colourings contribute h(e₁ ⊙ e₁) = θ and h(e₂ ⊙ e₂) = 0.

The value of the participating loop

The loop's two half-edges are matched to each other, so they form a single κ-circuit and the summand carries the sign (−1)¹. There are no even colours left, and the two odd colourings contribute through the same basis vector of Λ²V₁.

The evaluation

theorem RS.mixedPartition_loopGraph (θ : ℂ) : mixedPartition (charPolyFunctional θ) loopGraph = θ - 2

The paper's worked example: the mixed partition function of the functional h(θ) evaluates on the loop graph L to θ − 2, the characteristic polynomial det(θ I − A_L) of its adjacency matrix A_L = (2).

The two Eulerian subsets contribute θ and −2 respectively. The free-circle factor (k − 2ℓ)^circles is 0 ^ 0 = 1: the loop is an edge, not a free circle, and had it been one the value would have been 0.

def RS.loopGraphCircle : ClosedFragment

The loop graph with a free circle adjoined.

Equations

• One or more equations did not get rendered due to their size.

theorem RS.mixedPartition_loopGraphCircle (θ : ℂ) : mixedPartition (charPolyFunctional θ) loopGraphCircle = 0

A free circle annihilates. Here k − 2ℓ = 0, so the paper's (k − 2ℓ)^circles convention makes p_{h(θ)} vanish on every graph carrying a free circle. Set beside mixedPartition_loopGraph, this is the loop/free-circle distinction as a pair of numbers: the same one-vertex graph is worth θ − 2 when its edge is a loop and 0 when a circle rides alongside.