The fragment tensor #
RS21 attaches to a fragment, an Eulerian subset, an Eulerian orientation and a compatible local pairing the tensor
t′_h(F,H,ω,κ) := Σ_χ t′_{h,χ}(F,H,ω,κ),
t′_{h,χ} := (−1)^{c-hat(κ)} Σ_{ψ ∼ χ₀, φ ∼ χ₁}
∏_{v ∈ V′(F)} h_v( … ) ⊗_{i ∈ [t]} c_{χ,ω,i}.
A basis coordinate of the tensor determines χ: an entering leg
carries f_{χ₁(i)}, so its coordinate is χ₁(i) itself, and a
leaving leg carries g_{χ₁(i)}, whose expansion is the partner
colour with the partner sign. So the sum over χ collapses, and
the coordinate at x is the colourings' sum read at untwist x,
weighted by the leaving legs' signs.
The tensor is zero at a coordinate whose parity pattern is not the
subset's, which is the condition that χ be consistent with S.
RS21's tensor t′_h, in coordinates.
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RS21's tensor at given arc directions. The chain
orientation fixes the vertex signs; the arc directions fix which
legs carry f and which carry g.
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The dual weight is a product of leg weights #
The dual basis's weight is a product over the labels of a factor that depends only on that leg's colour and its arc's direction, so it is the leg weight the Gram computation uses. At a label the subset does not use, the colour is even and the factor is one.
The arc direction as a function of the label, with the unused
labels reading false.
Equations
- F.legDir tl i = if h : W.boundaryFlag i ∈ F.boundaryFlags then tl ⟨i, h⟩ else false
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The dual weight is the product of the legs' weights.
The normalised tensor #
RS21 normalises by a fourth root of unity per two used legs and by the matching's sign:
t_h(F,H,ω,κ) := (−1)^{|S|/4} · sgn(M(ω,κ)) · t′_h(F,H,ω,κ).
Since |S| is only even, (−1)^{|S|/4} is a fourth root: i^{|S|/2}.
The sign is taken against the reference matching with arcs
(i₁,i₂),…, which is stdMatching on the used labels.
The labels the subset uses — RS21's S(H).
Equations
- F.UsedLabel = { i : α // W.boundaryFlag i ∈ F.boundaryFlags }
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The used labels are even in number: they are matched in pairs.
RS21's normalised tensor t_h, in coordinates.
Equations
- F.tFull h κ o x = Complex.I ^ (Fintype.card F.UsedLabel / 2) * ↑↑((RS.DirMatching.stdMatching ⋯).sgnRel (RS.cutMatching F κ o)) * F.tPrime h κ o x
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The used labels are even in number, for any directed matching on them.
RS21's normalised tensor at given arc directions. The directions enter twice: through the matching's sign and through the dual basis.
Equations
- F.tFullD h κ o M x = Complex.I ^ (Fintype.card (RS.UsedLab F) / 2) * ↑↑((RS.DirMatching.stdMatching ⋯).sgnRel M) * F.tPrimeD h κ o M.tail x
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The normalised tensor is the directed one at the chain orientation's own directions.
A disagreeing state carries no colouring. RS21 colours a
through-edge once, so a state whose two legs there disagree admits
no φ ∼ χ₁, and the tensor vanishes at it.
A disagreeing state carries no colouring, at the chain orientation's own directions.
RS21's t_h vanishes at a disagreeing state.
The normalised tensor vanishes at a disagreeing state.
The tensor's support #
The tensor vanishes unless the state's odd legs are exactly the labels the subset uses. On that support the number of odd legs is the number of used labels, and the legs whose arc leaves are half of them. These are the hypotheses RS21's leg count needs.
The tensor vanishes off its support.
The same, for the normalised tensor.
On the support the odd legs are the used labels.
The leg direction agrees with the matching's on the used labels.
The fragment's change of basis is the abstract one at its own leg directions.
Half the used legs, in the form the leg count needs: the legs whose arc leaves are half the odd ones.
The tensor vanishes off its support, read on the state itself rather than on its untwist.
The normalised tensor vanishes off its support.
The tensor over the core sum #
RS21's colouring sum runs over every edge of the subset. By the bridge it equals the sum over the core edges, which is the vertex sum the mixed partition function is built from. So the tensor is the vertex sum, weighted by the circuit sign and the dual basis.
The tensor is the vertex sum, weighted.
The tensor at given arc directions is the vertex sum, weighted.
The tensor in the Gram computation's terms: a fourth root and the matching's sign, the circuit sign, the legs' weights, and the vertex sum at the untwisted state. Every factor but the last is what RS21's sign bookkeeping handles; the last is what the colouring sums multiply.
The two fragments' vertex sums, paired #
RS21's right-hand side is the composed graph's summand, whose
colouring sum runs over V′(G) = V′(F₁) ⊔ V′(F₂). The object the
Gram pairing produces is the two fragments' vertex sums multiplied
at a shared interface state; naming it separates the sign
bookkeeping from the colouring correspondence.
The two fragments' vertex sums at a shared agreeing state. RS21 colours a through-edge once, so a state whose two legs there disagree carries no colouring at all and both tensors vanish at it. The pairing therefore sees only the agreeing states, and it is this value, not the bare product of vertex sums, that it computes.
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The agreeing value on its support is the two colouring sums.
The agreeing value vanishes off the first tensor's support.
The agreeing value vanishes where the first side disagrees.
The agreeing value vanishes where the second side disagrees.
The paired value is the two colouring sums, in RS21's own form. The agreement is not a condition imposed on top: it is the support of the colouring sum itself.
The Gram summand, per coordinate. At a coordinate the first tensor supports, the product of the two tensors against the form is the sign bookkeeping, the twists and leg weights the cancellation consumes, and the two vertex sums at the shared state.
RS21's (13), up to its sign bookkeeping. The Gram pairing of the two fragments' tensors is the two colouring sums, multiplied at a shared interface state and summed, times the two matchings' signs and circuit signs. What remains to identify it with the composed graph's summand is (14) and the colouring correspondence.
The vertex sum under a chain flip #
The colouring sum's own chain-flip ledger, read on the vertex sum.
The vertex sum under a chain flip.
The vertex sum ignores the through legs #
The core colouring constraint reaches only the core flags, and a through edge's legs are not among them. Flipping the state's odd colour to its partner at two such legs therefore leaves the vertex sum alone: the even constraint does not see an odd leg at all, and the odd constraint does not see a through one.
Flipping the state at two through legs leaves the vertex sum unchanged.
The tensor under a chain flip #
RS21: inverting a directed trail negates t′_h. The two ledgers
meet — the dual basis contributes dualSign at each chain end, the
colouring sum contributes the flipped state's own sign there — and
each pair is 1 where the trail leaves and -1 where it enters.
Exactly one end of a chain is its tail, so the product is -1.
The dual sign against the state's own sign.
The dual sign against the partner colour's sign.
The tensor changes sign under a chain flip — RS21's
t′_h(F,H,ω,κ) = -t′_h(F,H,ω′,κ′). The dual basis contributes the
chain ends' own signs and the colouring sum contributes the flipped
state's; each pair is 1 at the end the trail leaves and -1 at the
end it enters, and a trail leaves exactly one of its two ends.
Inverting an edge joining two labelled ends #
RS21's (12) inverts a directed trail. When the trail is a single
edge with both ends labelled there is no transition to invert, so
the whole effect falls on the dual basis: the two legs exchange f
and g. The colouring moves too — the edge's colour becomes its
partner — but that colour occurs at no vertex, so the vertex sum is
unchanged and the two weights differ by exactly one sign. This is
RS21's count of one arc and no pairings.
Inverting an edge joining two labelled ends negates the tensor.
RS21's (12) at the normalised tensor: inverting an edge
joining two labelled ends leaves t_h alone, because the matching's
sign and the tensor both change sign.
RS21's (12) at the normalised tensor, for a chain:
inverting a directed trail through the interior leaves t_h alone,
because the matching's sign and the tensor both change sign. The
orientation moves with the trail, which is what distinguishes this
from the case of an edge joining two labelled ends.
RS21's (12) for a chain, at every state. Off the tensor's support both sides vanish, so the invariance needs no hypothesis on the state.
The tensor does not see those edges' directions #
Since inverting such an edge leaves t_h alone, and any two
direction assignments on them differ by a set of such inversions,
the normalised tensor is the same for all of them. This is RS21's
"we may assume that ω₁, κ₁, ω₂, κ₂ are chosen so that the union is
Eulerian", for the half of the choice the chain orientation does not
already provide.
The normalised tensor is independent of the directions given to the edges joining two labelled ends.
A through edge's two labels carry partner colours #
The colouring the tensor sums over gives a through edge one colour, so in the twisted basis its two labels carry partner colours — which is what RS21's (12) reads at such an edge. It is not an extra hypothesis: it follows from the agreement the colouring forces.
At a through label the chord is the edge's other end.
A through edge's two labels carry partner colours.
Agreement is a condition on the state alone. At a through edge it says the two labels' colours are partners, and reading that does not need the arc directions: reversing the edge replaces both ends' colours by their partners at once.
The agreement does not read the arc directions.
RS21's step 1, the chain half #
The directions at the chain labels are the chain orientation's own,
and a chain flip reverses exactly one of their arcs at no cost to the
tensor. So the orientation can be chosen to give those labels any
directions the pairing allows — which is half of "we may assume that
ω₁, κ₁, ω₂, κ₂ are chosen so that the union is Eulerian".
The chain labels' directions can be chosen freely.
RS21's step 1. The arc directions can be given any values the pairing allows, at no cost to the tensor: the chain labels' by choosing the orientation, the through labels' outright.
The Eulerian position #
RS21's step 1 asks for directions making M(ω₁,κ₁) ∪ M(ω₂,κ₂)
Eulerian, and Lemma 11's repair supplies them. Since the tensor does
not read the directions beyond their pairing, they can be imposed on
both sides at once.
The two subsets' used labels, identified by the interface.
Equations
- F₁.usedLabInterfaceEquiv F₂ hused = Equiv.subtypeEquivRight hused
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The two subsets' arc directions can be put in Eulerian position, keeping the pairings.
RS21's (13), with the directions discharged. Step 1 supplies the Eulerian position and the tensor does not see it, so the pairing of the two fragments' tensors is the two colouring sums against the sign bookkeeping, with no hypothesis on the directions.
The fragment's tensor #
Summing the normalised tensors over the Eulerian subsets gives the fragment's own tensor, at the transition data each subset's canonical data provide. Any choice of data serves, by the invariance under reversing a trail.
The fragment's tensor: Σ_H t_h(F,H,ω_H,κ_H).
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