Neurology Mathematics Ideas124.pdf - e(i with all possible values of i to Next we note that by successively applying the functors G n,\u03ba the

Neurology Mathematics Ideas124.pdf - e(i with all possible...

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Next we note that by successively applying the functors tildewide G ( i ) n,κ with all possible values of i to the transparently labeled gluing circle S n,κ (C.2) we obtain a fillable disk D n,κ each of whose inner boundaries is a tadpole circle. This allows for the following description of the silent object ( S n,κ ), as defined according to (5.40): Lemma C.5. The silent object n,κ := ( S n,κ ) for any transparent gluing circle S n,κ can be recovered from the functors tildewide G ( i ) ℓ,κ with = 2 , 3 ,...,n and with ( i 1 ,i 2 ,...,i n ) any permutation of (1 , 2 ,...,n ) as follows (for brevity we abuse notation by writing the same generic label κ for all the tuples of framing indices involved): there is a canonical isomorphism ρ : n,κ = −−→ tildewide G ( ( Q ǫ i 1 )) with tildewide G the composite T( Q ǫ i 1 ) tildewide G ( i 2 ) 2 −−−−→ T( S 2 ) tildewide G ( i 3 ) 3 −−−−→ T( S 3 ) tildewide G ( i 4 ) 4 −−−−→ · · · · · · tildewide G ( in ) n,κ −−−−→ T( S n,κ ) . (C.23)
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