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.

The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another. Owing to the difficulty in computing higher co-homology operations, one is led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. / Science, Faculty of / Mathematics, Department of / Graduate

The graded cohomology groups of a cartesian product of two cellular spaces are expressible in terms of the cohomology groups of the factors. This relationship is given by the (split) short exact Runneth sequence. However the multiplicative structure on the cohomology of a cartesian product can in general not be derived by solely referring to the Runneth formula. In this thesis we explicitly exhibit the cup product structure on a cartesian product of two (standard) lens spaces. This result is obtained by analyzing the Runneth sequence and by making use of the particular geometry of the spaces involved. / Science, Faculty of / Mathematics, Department of / Graduate

Symplectic Khovanov homology is first defined by Seidel and Smith as a singly graded link homology. It is proved isomorphic to combinatorial Khovanov homology over any characteristic zero field by Abouzaid and Smith. In this dissertation, we construct a second grading on symplectic Khovanov homology from counting holomorphic disks in a partially compactified space. One of the main theorems asserts that this grading is well-defined. We also conclude the other main theorem that this second grading recovers the Jones grading of Khovanov homology over any characteristic zero field, through showing that the Abouzaid and Smith's isomorphism can be refined as an isomorphism between doubly graded groups. The proof of the theorem is carried out by showing that there exists a long exact sequence in symplectic Khovanov homology that commutes with its combinatorial counterpart.

La théorie de l'homologie fournit des solutions nouvelles et efficaces pour régler le problème de trou de couverture dans les réseaux de capteurs sans fil. Ils sont basés sur deux objets combinatoires nommés complexe de Cech et complexe de Rips. Le complexe de Cech peut détecter l'intégralité des trous de couverture, mais il est très difficile à construire. Le complexe de Rips est facile à construire, mais il est imprécis dans certaines situations. Dans la première partie de cette thèse, nous choisissons la proportion de la surface de trous manqués par le complexe de Rips comme une mesure d'évaluer l'exactitude de la détection de trou de couverture basée sur l'homologie. Des expressions fermées pour les bornes inférieures et supérieures de la proportion sont dérivés. Les résultats de simulation sont bien compatibles avec les bornes inférieure et supérieure calculés analytiquement, avec des différences maximales de 0.5% et 3%. En outre, nous étendons l'analyse au cas de la sphère. Dans la deuxième partie, nous proposons d'abord un algorithme distribué basé sur les graphes pour détecter les trous non triangulaires. Cet algorithme présente une grande complexité. Nous proposons donc un autre algorithme distribué plus efficace basé sur l'homologie. Cet algorithme ne nécessite que des informations de 1- et 2-saut nœuds voisins et a la complexité O(n3) où n est le nombre maximum de nœuds voisins à 1 saut. Il peut détecter avec précision les cycles frontières d'environ 99% des trous de couverture dans environ 99% des cas. / Homology theory provides new and powerful solutions to address the coverage hole problem in wireless sensor networks (WSNs). They are based on two combinatorial objects named Cech complex and Rips complex. Cech complex can fully characterize coverage properties of a WSN (existence and locations of holes), but it is very difficult to construct. Rips complex is easy to construct but it may miss some coverage holes. In the first part of this thesis, we choose the proportion of the area of holes missed by Rips complex as a metric to evaluate the accuracy of homology based coverage hole detection. Closed form expressions for lower and upper bounds of the proportion are derived. Simulation results are well consistent with the analytical lower and upper bounds, with maximum differences of 0.5% and 3%. In addition, we extend the analysis to the sphere case. In the second part, we first propose a graph based distributed algorithm to detect non-triangular holes. This algorithm exhibits high complexity. We thus propose another efficient homology based distributed algorithm. This algorithm only requires 1- and 2-hop neighbour nodes information and has the worst case complexity O(n3) where n is the maximum number of 1-hop neighbour nodes. It can accurately detect the boundary cycles of about 99% coverage holes in about 99% cases.

Despite the biological importance of lepidopteran wing patterns, homologies between pattern elements in different lineages are still not understood. Though plesiomorphic wing veins influence color patterning even when not expressed in the adult wing, most studies of wing pattern evolution have focused on derived taxa with reduced venation. Here I address this gap with an examination of Micropterigidae, a very early-diverged family in which all known plesiomorphic lepidopteran veins are expressed in the adult wing. Differences between the coloration of transverse bands in Micropterix and Sabatinca suggest that homologies exist between the contrast boundaries that divide wing pattern elements. Because the wing pattern of Sabatinca doroxena very closely resembles the nymphalid groundplan when plotted onto a hypothetical nymphalid wing following the relationship between pattern and venation discussed here, it appears that the nymphalid groundplan may have originated from a Sabatinca-like wing pattern subjected to changes in wing shape.

Thesis advisor: John Baldwin / Heegaard Floer homology is a family of invariants in low dimensional topology due originally to Ozsváth-Szabó. We discuss various aspects of Heegaard Floer homology and give several link detection results for versions of Heegaard Floer homology for links. In particular, we show that knot and link Floer homology detect various infinite families of cable links. We also give classification results for the Heegaard Floer theoretic invariants of a type of knot called an “almost L-space knot” and an infinite family of detection results for annular Khovanov homology. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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