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Subsections
3.4 Integrability conditions or
Figure 12:
Undoing a Bundle Left Twist one crossing at a time.

Figure 13:
The Total Twist Relation (TTR).

Figure 14:
The Total Twist Relation (TTR).

Figure 15:
The Monster

Figure 16:
Lassoing a Borromean link.

(The last equality holds because in the two error terms,
Y_{rab}Y_{rgb} and Y_{rgb}, the component p is unknotted).
Now reduce the component r using the total twist relation. Only the first
term is affected, and 3 of the 6 terms that are produced from its
reduction cancel against the 3 remaining terms of the above equation. The
result is:
The last term here drops out because in it the component r is unknotted,
and so the end result is
.
In graphical terms,
this is precisely the graph I! Cyclically permuting the roles of r,
g, and b, we find that we have proven the IHX relation.
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Dror BarNatan
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