<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> / Dissertation

We study the sl(3) web algebra via morphisms on foams. A pre-foam is a cobordism between two webs that contains singular arcs, which are sets of points whose neighborhoods are homeomorphic to the cross-product of the letter "Y'' and the unit interval. Pre-foams may have a distinguished point, and it can be moved around as long as it does not cross a singular arc. A foam is an isotopy class of pre-foams modulo a set of certain relations involving dots on the pre-foams. Composition in Foams is achieved by stacking pre-foams. We compute the cohomology ring of the sl(3) web algebra and apply a functor from the cohomology ring of the sl(3) web algebra to {\bf Foams}. Afterwards, we use this to study the $\mathfrak{sl}(3)$ web algebra via morphisms on foams.

Thesis advisor: Julia Elisenda Grigsby / In 1999, Khovanov constructed a combinatorial categorification of the Jones polynomial. Since then there has been a question of to what extent the topology of a link is reflected in his homology theory and how Khovanov homology can be used for topological applications. This dissertation compiles some of the authors contributions to these avenues of mathematical inquiry. In the first chapter, we prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen $s$-invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $\psi$ invariant is entirely determined by the $s$-invariant and the self-linking number for 3-braid closures. In the second chapter, we show if $L$ is any link in $S^3$ whose Khovanov homology is isomorphic to the Khovanov homology of $T(2,6)$ then $L$ is isotopic to $T(2,6)$. We show this for unreduced Khovanov homology with $\mathbb{Z}$ coefficients. Finally in the third chapter, we exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

A homology theory is defined for equivalence classes of links under isotopy in the 3-sphere. Chain modules for a link L are generated by certain surfaces whose boundary is L, using surface signature as the homological grading. In the end, the diagramless homology of a link is found to be equal to some number of copies of the Khovanov homology of that link. There is also a discussion of how one would generalize the diagramless homology theory (hence the theory of Khovanov homology) to links in arbitrary closed oriented 3-manifolds.