Are isotopic PL links PL isotopic? (part III)

This part of the talk will be mostly independent of the previous parts. We are going to review and apply the clasper theory of M. Gusarov and K. Habiro.
Namely, we will start by revisiting G. Massuyeau's theorem that string links that are not separated by type $n$ invariants are $C_k$-equivalent for some $k=k(n)$. We are going to reprove this theorem by correcting Habiro's earlier attempt to prove its weaker version (with $k$ depending not only on $n$ but also on the two string links). Although the main geometric lemma will be quoted without proof from Habiro's paper, the reason why $k$ may have to differ from $n$ will be fully explained.
As an application of this technique, we will show that all finite type invariants of string links contain the same information as locally additive ones (=additive under insertion of local knots). This is the remaining ingredient needed to complete the proof of the main theorem (that topologically isotopic PL links are PL isotopic to links which are not separated by finite type invariants).

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