This paper explains the construction of Khovanov homology of which begins by un derstanding how Louis Kauffman generalizes the Jones polynomial using a state sum model of the bracket polynomial for an unoriented knot or link and in turn recovers the Jones polynomial, a knot invariant for an oriented knot or link. Kauffman associates the unknot by the polynomial (−A2 − A−2) whereas Khovanov associates the unknot by (q + q−1) through a change of variables. As an oriented knot or link K with n crossings produces 2n smoothings, Khovanov builds a commutative cube {0,1}n and associates a graded vector space to each smoothing in the cube. By defining a differential operator on the directed edges of the cube so that adjacent states differ by a type of smoothing for a fixed cross ing, we can form chain groups which are direct sums of these vector spaces. Naturally we get a bi-graded (co)chain complex which is called the Khovanov complex. The resulting (co)homology groups of these (co)chains turns out to be invariant under the Reidemeister moves and taking the Euler characteristic of the Khovanov complex returns the very same Jones polynomial that we started with.

Thesis advisor: Julia Elisenda Grigsby / Thesis advisor: David Treumann / In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: John A. Baldwin / Khovanov homology and Heegaard Floer homology have opened new horizons in knot theory and three-manifold topology, respectively. The two invariants have distinct origins, but the Khovanov homology of a link is related to the Heegaard Floer homology of its branched double cover by a spectral sequence constructed by Ozsváth and Szabó. In this thesis, we construct an equivalent spectral sequence with a much more transparent connection to Khovanov homology. This is the first step towards proving Seed and Szabó's conjecture that Szabó's geometric spectral sequence is isomorphic to Ozsváth and Szabó's spectral sequence. These spectral sequences connect information about contact structures contained in each invariant. We construct a braid conjugacy class invariant κ from Khovanov homology by adapting Floer-theoretic tools. There is a related transverse invariant which we conjecture to be effective. The conjugacy class invariant solves the word problem in the braid group among other applications. We have written a computer program to compute the invariant. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

The Khovanov{Lauda{Rouquier algebras Rn are a relatively new family of Z-graded algebras. Their cyclotomic quotients R n are intimately connected to a smaller family of algebras, the cyclotomic Hecke algebras H n of type A, via Brundan and Kleshchev's Graded Isomorphism Theorem. The study of representation theory of H n is well developed, partly inspired by the remaining open questions about the modular representations of the symmetric group Sn. There is a profound interplay between the representations for Sn and combinatorics, whereby each irreducible representation in characteristic zero can be realised as a Specht module whose basis is constructed from combinatorial objects. For R n , we can similarly construct their representations as analogous Specht modules in a combinatorial fashion. Many results can be lifted through the Graded Isomorphism Theorem from the symmetric group algebras, and more so from H n , to the cyclotomic Khovanov{Lauda{Rouquier algebras, providing a foundation for the representation theory of R n . Following the introduction of R n , Brundan, Kleshchev and Wang discovered that Specht modules over R n have Z-graded bases, giving rise to the study of graded Specht modules. In this thesis we solely study graded Specht modules and their irreducible quotients for R n . One of the main problems in graded representation theory of R n , the Graded Decomposition Number Problem, is to determine the graded multiplicities of graded irreducible R n -modules arising as graded composition factors of graded Specht modules. We rst consider R n in level one, which is isomorphic to the Iwahori{Hecke algebra of type A, and research graded Specht modules labelled by hook partitions in this context. In quantum characteristic two, we extend to R n a result of Murphy for the symmetric groups, determining graded ltrations of Specht modules labelled by hook partitions, whose factors appear as Specht modules labelled by two-part partitions. In quantum characteristic at least three, we determine an analogous R n -version of Peel's Theorem for the symmetric groups, providing an alternative approach to Chuang, Miyachi and Tan. We then study graded Specht modules labelled by hook bipartitions for R n in level two, which is isomorphic to the Iwahori{Hecke algebra of type B. In quantum characterisitic at least three, we completely determine the composition factors of Specht modules labelled by hook bipartitions for R n , together with their graded analogues.

Cette thèse établit, et étudie, un lien entre l'homologie de Khovanov et la topologie des revêtements ramifiés doubles. Nous y introduisons certaines propriétés de stabilité en homologie de Khovanov, dont nous dérivons par la suite des obstructions à l'existence de certaines chirurgies exceptionnelles sur les noeuds admettant une involution appropriée. Ce comportement, analogue à celui de l'homologie de Heegaard-Floer sous chirurgie, renforce ainsi le lien existant (dû à Ozsváth et Szabó) entre homologie de Khovanov, et homologie d'Heegaard-Floer des revêtements ramifiés doubles. Dans l'optique de poursuivre et d'exploiter plus avant cette relation, les méthodes développées dans ce travail sont appliquées à l'étude des L-espaces, et à déterminer, en premier lieu, si l'homologie de Khovanov fournit un invariant des revêtements ramifiés doubles, et en deuxième lieu, si l'homologie de Khovanov permet de détecter le noeud trivial. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Homologie de Khovanov, Homologie de Heegaard-Floer, Chirurgies de Dehn, Involutions, Variétés de dimension trois, Revêtements ramifiés doubles.

Knot theory is a rich topic in topology that studies the how circles can be embedded in Euclidean 3-space. One of the main questions in knot theory is how to distinguish between different types of knots efficiently. One way to approach this problem is to study knot invariants, which are properties of knots that do not change under a standard set of deformations. We give a brief overview of basic knot theory, and examine a specific knot invariant known as Khovanov homology. Khovanov homology is a homological invariant that refines the Jones polynomial, another knot invariant that assigns a Laurent polynomial to a knot. Dror Bar-Natan wrote a paper in 2002 that explains the construction of Khovanov homology and proves that it is an invariant. We follow his lead and attempt to clarify and explain his formulation in more precise detail.

Thesis advisor: Julia E. Grigsby / The first part of this thesis is on properties of annular Khovanov homology. We prove a connection between the Euler characteristic of annular Khovanov homology and the classical Burau representation for closed braids. This yields a straightforward method for distinguishing, in some cases, the annular Khovanov homologies of two closed braids. As a corollary, we obtain the main result of the first project: that annular Khovanov homology is not invariant under a certain type of mutation on closed braids that we call axis-preserving. The second project is joint work with Adam Saltz. Plamenevskaya showed in 2006 that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this project we define an annular refinement of this element, kappa, and show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We first discuss examples that show that kappa is non-trivial. We then prove applications of kappa relating to braid stabilization and spectral sequences, and we prove that kappa provides a new solution to the word problem in the braid group. Finally, we discuss definitions and properties of kappa in the reduced setting. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Julia E. Grigsby / We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element δ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of Bₙ and we prove that Rudolph’s condition is equivalent to ours. We compute the sutured Khovanov homology groups of positive braid closures in homological degrees i = 0,1 as sl₂(ℂ)-modules. Given a condition on the sutured Khovanov homology of strongly quasipositive braids, we show that the sutured Khovanov homology of the closure of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element agrees with that of positive braid closures in homological degrees i ≤ 1 and show this holds for the class of such braids on three strands. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.

Knots and links $L\subset S^3$ carry a wealth of data. Spanning surfaces $F$ (1- or 2-sided), $\partial F=L$, especially {\bf checkerboard} surfaces from link diagrams $D\subset S^2$, help to mine this data. This text explores the structure of these surfaces, with a focus on a gluing operation called {\bf plumbing}, or {\it Murasugi sum}. First, naive classification questions provide natural and accessible motivation for the geometric and algebraic notions of essentiality (incompressibility with $\partial$-incompressibility and $\pi_1$-injectivity, respectively). This opening narrative also scaffolds a system of hyperlinks to the usual background information, which lies out of the way in appendices and glossaries. We then extend both notions of essentiality to define geometric and algebraic {\it degrees} of essentiality, $\underset{\hookrightarrow}{\text{ess}}(F)$ and $\text{ess}(F)$. For the latter, cutting $S^3$ along $F$ and letting $\mathcal{X}$ denote the set of compressing disks for $\partial (S^3\backslash\backslash F)$ in $S^3\backslash\backslash F$, $\text{ess}(F):=\min_{X\in\mathcal{X}}|\partial X\cap L|$. Extending results of Gabai and Ozawa, we prove that plumbing respects degrees of algebraic essentiality, $\text{ess}(F_1*F_2)\geq\min_{i=1,2}\text{ess}(F_i)$, provided $F_1,F_2$ are essential. We also show by example that plumbing does not respect the condition of geometric essentiality. We ask which surfaces de-plumb uniquely. We show that, in general, essentiality is necessary but insufficient, and we give various sufficient conditions. We consider Ozawa's notion of representativity $r(F,L)$, which is defined similarly to $\text{ess}(F)$, except that $F$ is a closed surface in $S^3$ that contains $L$, rather than a surface whose boundary equals $L$. We use Menasco's crossing bubbles to describe a sort of thin position for such a closed surface, relative to a given link diagram, and we prove in the case of alternating links that $r(F,L)\leq2$. (The contents of Chapter 4, under the title Alternating links have representativity 2, are first published in Algebraic \& Geometric Topology in 2018, published by Mathematical Sciences Publishers.) We then adapt these arguments to the context of spanning surfaces, obtaining a simpler proof of a useful crossing band lemma, as well as a foundation for future attempts to better classify the spanning surfaces for a given alternating link. We adapt the operation of plumbing to the context of Khovanov homology. We prove that every homogeneously adequate Kauffman state has enhancements $X^\pm$ in distinct $j$-gradings whose traces (which we define) represent nonzero Khovanov homology classes over $\mathbb{Z}/2\mathbb{Z}$, and that this is also true over $\mathbb{Z}$ when all $A$-blocks' state surfaces are two-sided. A direct proof constructs $X^\pm$ explicitly. An alternate proof, reflecting the theorem's geometric motivation, applies our adapted plumbing operation. Finally, we describe an interpretation of Khovanov homology in terms of decorated cell decompositions of abstract, nonorientable surfaces, featuring properly embedded (1+1)-dimensional nonorientable cobordisms in (2+1)-dimensional nonorientable cobordisms. This formulation contains a planarity condition; removing this condition leads to Khovanov homology for virtual link diagrams.