4.3 Structural approximations
For a type index \( \beta \), a \( \beta \)-approximation is a \( \beta \)-tree of base approximations. We define the partial order on \( \beta \)-approximations branchwise. We define an action of \( \beta \)-approximations \( \psi \) on \( \beta \)-supports \( S \) by \( (\psi (S))_A = \psi _A(S_A) \).
Let \( A \) be a \( \beta \)-extended type index. A litter \( L \) is \( A \)-inflexible if there is an inflexible \( \beta \)-path \( I \) such that \( A = ((A_I)_{\varepsilon _I})_\bot \) and \( L = f_{\delta _I, \varepsilon _I}(t) \) for some \( t : \mathsf{Tang}_{\delta _I} \). The coderivative operation works in the obvious way. A litter can be \( A \)-inflexible in at most one way. 1
We say that a \( L \) is \( A \)-flexible if it is not \( A \)-inflexible. 2 If \( L \) is \( B_A \)-flexible, then \( L \) is \( A \)-flexible.
A \( \beta \)-approximation \( \psi \) is coherent at \( (A, L_1, L_2) \) if:
If \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \), then there is some \( \delta \)-allowable permutation \( \rho \) such that
\[ (\psi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \]and
\[ L_2 = f_{\delta ,\varepsilon }(\rho (t)) \](and hence all \( \delta \)-allowable permutations \( \rho \) such that \( (\psi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \) satisfy \( L_2 = f_{\delta ,\varepsilon }(\rho (t)) \)).
If \( L_1 \) is \( A \)-flexible, then \( L_2 \) is \( A \)-flexible.
We say that \( \psi \) is coherent if whenever \( (L_1, L_2) \in \psi _A^{\mathcal L}\), \( \psi \) is coherent at \( (A, L_1, L_2) \).
Suppose that \( \psi \) is an approximation and \( L : \mathbb Z \to {\mathcal L}\) is a function satisfying the hypotheses of proposition 4.11. Let \( \chi \) be the extension produced by the structural version of this result. 3 If \( \psi \) is coherent, and is additionally coherent at \( (L(n), L(n+1)) \) for each integer \( n \), then \( \chi \) is coherent.
This proof just relies on the fact that if \( (\psi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \), then the same holds for every extension \( \chi \) of \( \psi \). 4
If \( \psi \) is coherent, then \( \psi ^{-1} \) is coherent.
Suppose that \( (L_1, L_2) \in (\psi ^{-1}_A)^{\mathcal L}\), so \( (L_2, L_1) \in \psi _A^{\mathcal L}\). Suppose first that \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). If \( L_2 \) were \( A \)-flexible, then \( L_1 \) would also be \( A \)-flexible by coherence, which is a contradiction. So \( L_2 \) is \( A \)-inflexible with path \( (\gamma ', \delta ', \varepsilon ', B') \) and tangle \( t' : \mathsf{Tang}_{\delta '} \), and there is \( \rho : \mathsf{AllPerm}_{\delta '} \) such that
and
We thus deduce \( \varepsilon = \varepsilon ' \) and \( \gamma = \gamma ' \) by the equations for \( A \). By the equation \( L_1 = f_{\delta ,\varepsilon }(t) \), we also obtain \( \delta = \delta ' \) and \( t = \rho (t') \). Then we find
where the last equation uses the fact that \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t') \). Finally, the equation \( L_2 = f_{\delta ,\varepsilon }(\rho ^{-1}(t)) \) gives coherence at \( (A, L_1, L_2) \) as required.
Now suppose that \( L_1 \) is \( A \)-flexible. If \( L_2 \) were \( A \)-inflexible, then so would be \( L_1 \) by coherence. So \( L_2 \) is \( A \)-flexible, as required.
If \( \psi \) and \( \chi \) are coherent and have equal coimages along all paths, then \( \psi \circ \chi \) is coherent.
Suppose that \( (L_1, L_3) \in ((\psi \circ \chi )_A)^{\mathcal L}\), so \( (L_1, L_2) \in \psi _A^{\mathcal L}\) and \( (L_2, L_3) \in \chi _A^{\mathcal L}\). Suppose that \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). Then by coherence of \( \psi \), we have \( \rho \) such that
and
Then \( L_2 \) is \( A \)-inflexible with path \( I \) and tangle \( \rho (t) \). So by coherence of \( \chi \), we have \( \rho ' \) such that
and
As \( \rho '(\operatorname {\mathsf{supp}}(\rho (t))) = \rho '(\rho (\operatorname {\mathsf{supp}}(t))) \), we obtain the desired coherence result.
Instead, if \( L_1 \) is \( A \)-flexible, then so is \( L_2 \) by coherence of \( \psi \), and so is \( L_3 \) by coherence of \( \chi \).
If \( \psi \) is a coherent \( \beta \)-approximation and \( A \) is a path \( \beta \rightsquigarrow \beta ' \), then \( \psi _A \) is a coherent \( \beta ' \)-approximation.
Let \( (L_1, L_2) \in (\psi _A)_B^{\mathcal L}\). Suppose that \( L_1 \) is \( B \)-inflexible with path \( (\gamma , \delta , \varepsilon , C) \) and \( t : \mathsf{Tang}_\delta \). Then \( L_1 \) is \( A_B \)-inflexible with path \( (\gamma , \delta , \varepsilon , A_C) \) and the same tangle \( t \). So by coherence of \( \psi \), we obtain a \( \delta \)-allowable \( \rho \) such that
and
This same \( \rho \) can thus be used to establish coherence of \( \psi _A \) at \( (B, L_1, L_2) \).
Thus, by proposition 4.16, whenever \( L_2 \) is \( B \)-inflexible with path \( I \) and tangle \( t \), \( L_1 \) is also \( B \)-inflexible with path \( I \). So if \( L_1 \) is \( B \)-flexible, so is \( L_2 \), as required.