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<p>Quantum sl_3 projectors are morphisms in Kuperberg's sl_3 spider, a diagrammatically
defined category equivalent to the full pivotal subcategory of the category of (type
1) finite-dimensional representations of the quantum group U_q (sl_3 ) generated by
the defining representation, which correspond to projection onto (and then inclusion
from) the highest weight irreducible summand. These morphisms are interesting from
a topological viewpoint as they allow the combinatorial formulation of the sl_3 tangle
invariant (in which tangle components are labelled by the defining representation)
to be extended to a combinatorial formulation of the invariant in which components
are labelled by arbitrary finite-dimensional irreducible representations. They also
allow for a combinatorial description of the SU(3) Witten-Reshetikhin-Turaev 3-manifold
invariant. </p><p>There exists a categorification of the sl_3 spider, due to Morrison
and Nieh, which is the natural setting for Khovanov's sl_3 link homology theory and
its extension to tangles. An obvious question is whether there exist objects in this
categorification which categorify the sl_3 projectors. </p><p>In this dissertation,
we show that there indeed exist such "categorified projectors," constructing them
as the stable limit of the complexes assigned to k-twist torus braids (suitably shifted).
These complexes satisfy categorified versions of the defining relations of the (decategorified)
sl_3 projectors and map to them upon taking the Grothendieck group. We use these categorified
projectors to extend sl_3 Khovanov homology to a homology theory for framed links
with components labeled by arbitrary finite-dimensional irreducible representations
of sl_3 .</p>
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