To see the other types of publications on this topic, follow the link: Homologie de Khovanov.
Dissertations / Theses on the topic 'Homologie de Khovanov'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 33 dissertations / theses for your research on the topic 'Homologie de Khovanov.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Wagner, Emmanuel. "On Khovanov-Rozansky homology of graphs and links." Université Louis Pasteur (Strasbourg) (1971-2008), 2007. https://publication-theses.unistra.fr/restreint/theses_doctorat/2007/WAGNER_Emmanuel_2007.pdf.
Full textAbstract:
Cette thèse est consacrée à la catégorification d'invariants polynomiaux d'entrelacs et de graphes. Pour tout entier strictement positif n, Khovanov et Rozansky ont introduit en 2004 une homologie bigraduée d'entrelacs, ainsi qu'une homologie de graphes planaires. Etant donné n, leur homologie d'entrelacs catégorifie la n-ième spécialisation du polynôme d'entrelacs HOMFLYPT et leur homologie de graphes planaires catégorifie un polynôme de graphes associé. Dans cette thèse, on étudie ces homologies et on généralise leur construction en introduisant une graduation supplémentaire. Tout d'abord, on généralise une formule de Jaeger pour les polynômes d'entrelacs aux polynômes de graphes planaires, ainsi qu'à l'homologie de graphes planaires; on étend ensuite l'homologie d'entrelacs de Khovanov-Rozansky aux graphes plongés. Puis on construit une homologie trigraduée d'entrelacs. Cette homologie recouvre l'homologie bigraduée d'entrelacs de Khovanov et Rozansky. Enfin, on donne des exemples, des applications et des généralisations de l'homologie trigraduée d'entrelacs. On développe des outils d'algèbre homologique qui permettent de calculer explicitement l'homologie trigraduée d'entrelacs pour des exemples et on considère des déformations de l'homologie trigraduée d'entrelacsThis thesis is devoted to the categorification of polynomial invariants of graphs and links. For any positive integer n, Khovanov and Rozansky introduced in 2004 a bigraded link homology, and an homology of planar graphs. Given n, their link homology categorifies the n-th specialization of the HOMFLY-PT polynomial and their homology of planar graphs categorifies an associated graph polynomial. In this thesis, we study these homology and generalize their constructions by introducing an additional grading. First, we generalize a formula of Jaeger for link polynomials to polynomials of planar graphs and associated homology of planar graphs; we extend also the link homology of Khovanov and Rozansky to embedded graphs. Then we construct a triply graded link homology. This homology recovers the bigraded link homology of Khovanov and Rozansky. Finally, we give examples, applications and generalizations of the triply graded link homology. We develop homological tools that permit to compute explicitly the triply graded link homology for some knots and we consider deformations of the triply graded link homology
2
Wagner, Emmanuel Touraev Vladimir G. "On Khovanov-Rozansky homology of graphs and links." Strasbourg : Université Louis Pasteur, 2008. http://eprints-scd-ulp.u-strasbg.fr:8080/00000912.
Full text
3
Lewark, Lukas. "Homologies de Khovanov-Rozansky, toiles nouées pondérées et genre lisse." Paris 7, 2013. http://www.theses.fr/2013PA077117.
Full textAbstract:
Cette thèse porte sur les homologies de Khovanov-Rozansky et les invariants de concordance des nœuds qui en proviennent, en prêtant une attention particulière à l'homologie s13 définie par des mousses. Le premier chapitre est consacré aux interdépendances des différentes homologies de Khovanov-Rozansky : les homologies non-réduite et réduite, graduée et filtrée, et les homologies Homflypt et slN pour différents valeurs de N. Grâce à une composition des suites spectrales connues et nouvelles, on démontre sur des exemples que les invariants de concordance slN ne sont pas tous égaux ; ce résultat constitue une réponse à un problème ouvert jusqu'à ici. Le deuxième et troisième chapitres présentent une implémentation d'un algorithme qui calcule l'homologie s!3. Hormis le programme de Bar-Natan, Green et Morrison, donnant l'homologie de Khovanov, il s'agit du seul programme pour calculer une des homologies de Khovanov-Rozansky d'une manière efficace. Les calculs démontrent que l'invariant de concordance s13 peut prendre des valeurs impaires. Dans le quatrième chapitre, les homologies s!3 graduées et filtrées sont étendues à une classe des graphes noués et F3- pondérés : les toiles nouées pondérées. Les mousses pondérables, qui jouent le rôle des cobordismes orientables pour les toiles pondérées, permettent de définir la notion de degré lisse pour des toiles nouées pondérées. Par analogie avec le travail de Rasmussen, on démontre qu'une borne inférieure au degré lisse des toiles nouées pondérées découle de l'homologie s13 filtréeThis thesis focuses on the Khovanov-Rozansky homologies and the knot concordance invariants issuing from them, paying particular attention to the s13-foam homology. The first chapter treats the interrelation of different Khovanov-Rozansky homologies: unreduced and reduced, graded and filtered, and categorifying the Homflypt-polynomial and the slN-polynomial for varying N. A combination of new and known spectral sequences allows to show exemplarily that the slN-knot concordance invariants may differ, which was unknown until now. In the second and third chapter, an implementation of an algorithm Computing s13-homology is presented. Aside from Bar-Natan, Green and Morrisons' programme calculating Khovanov homology, this is the only existing programme that efficiently computes any Khovanov-Rozansky homology theory. Its calculations show that the s!3-knot concordance invariant may be an odd integer. In the fourth chapter, graded and filtered s!3-homology are generalised to a class of knotted F3-weighted graphs, called knotted weighted webs. Weightable foams are defined, which are to knotted weighted webs what orientable cobordisms are to knots, and the slice degree of knotted weighted webs is introduced. In analogy with Rasmussen's result, it is shown that the filtered sl3-homology yields a lower bound for the slice degree of knotted weighted webs
4
Robert, Louis-Hadrien. "Sur l'homologie sl3 des enchevêtrements : algèbres de Khovanov - Kuperberg." Paris 7, 2013. http://www.theses.fr/2013PA077240.
Full textAbstract:
Cette thèse est consacrée aux algèbres de Khovanov-Kuperberg Kc et à leurs catégories de modules. Ce sont les analogues dans le cas s13 , des algèbres H n utilisées par Khovanov pour étendre l'homologie s12 (ou homologie de Khovanov) aux enchevêtrements. Elles apparaissent comme les images des 0-objets par une (0+1+1)-TQFT. Elles permettent de définir une homologie s13 aux enchevêtrements. Ces algèbres ont des catégories de modules particulièrement intéressantes : du fait même de leurs constructions, elles sont profondément liées à l'étude des bases de certaines représentations du groupe quantique Uq(s13 ). Il est alors naturel de vouloir classifier les modules projectifs indécomposables sur ces algèbres. Nous étudions les modules de toiles qui sont des Kr -modules projectifs. Il a été conjecturé que ces modules formaient une famille complète de représentants des classes d'isomorphismes de Kc-modules projectifs indécomposables, mais Khovanov et Kuperberg ont exhibé un module de toile qui se décompose. Dans cette thèse nous donnons deux condition sur l'indécomposabilité des modules de toiles : une condition suffisante de nature géométrique, et une condition nécessaire et suffisante de nature algébrique. Les résultats sont prouvés d'une part grâce à une étude combinatoire des toiles, qui sont des graphes trivalents bipartites plan et d'un polynôme de Laurent qui leur est associé, le Crochet de Kuperberg. Et d'autre part, grâce à l'étude des mousses qui jouent le rôle de cobordismes pour les toilesThis thesis is concerned with the Khovanov-Kuperberg algebras Ke and to their categories of modules. They are analogous, in the s13 context, to the H n algebras used by Khovanov to extend the s12 -homology (or Khovanov homology) to tangles. They appear as images of the 0-objects by a (0+1+1)-TQFT. They allow to define a s13 -homology for tangles. The categories of modules over: these algebras are especially interesting: from their own construction they are deeply connected to bases in some representations of the quantum group Uq(s13). It is hence natural to ask for a classification of the projective indecomposable modules over these algebras. We study web modules which are projective Kc -modules. It has been conjectured that these modules constitute a complete family of indecomposable projective KE -modules, but Khovanov and Kuperberg have exhibited a web module which decomposes as a direct sum. In this thesis we give two conditions on the indecomposability of web modules: a geometric sufficient condition and an algebraic necessary and sufficient condition. The results are proven on the one hand, through a combinatorial analysis of webs which are plane bicubic graphs, and of Laurent polynomial associated with each web called the Kuperberg bracket. And on the other hand, thanks to foams which plays the role of cobordisms for webs
5
Wagner, Emmanuel. "Sur l'homologie de Khovanov-Rozansky des graphes et des entrelacs." Phd thesis, Université Louis Pasteur - Strasbourg I, 2007. http://tel.archives-ouvertes.fr/tel-00192447.
Full textAbstract:
Cette thèse est consacrée à la catégorification d'invariants polynomiaux d'entrelacs et de graphes. Pour tout entier strictement positif n, Khovanov et Rozansky ont introduit en 2004 une homologie bigraduée d'entrelacs, ainsi qu'une homologie de graphes planaires. Etant donné n, leur homologie d'entrelacs catégorifie la n-ième spécialisation du polynôme d'entrelacs HOMFLYPT et leur homologie de graphes planaires catégorifie un polynôme de graphes associé. Dans cette thèse, on étudie ces homologies et on généralise leur construction en introduisant une graduation supplémentaire. Tout d'abord, on généralise une formule de Jaeger pour les polynômes d'entrelacs aux polynômes de graphes planaires, ainsi qu'à l'homologie de graphes planaires; on étend ensuite l'homologie d'entrelacs de Khovanov-Rozansky aux graphes plongés. Puis on construit une homologie trigraduée d'entrelacs. Cette homologie recouvre l'homologie bigraduée d'entrelacs de Khovanov et Rozansky. Enfin, on donne des exemples, des applications et des généralisations de l'homologie trigraduée d'entrelacs. On développe des outils d'algèbre homologique qui permettent de calculer explicitement l'homologie trigraduée d'entrelacs pour des exemples et on considère des déformations de l'homologie trigraduée d'entrelacs.
6
Wagner, Emmanuel Turaev Vladimir G. "Sur l'homologie de Khovanov-Rozansky des graphes et des entrelacs /." Paris, 2007. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=016808065&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
Full text
7
Audoux, Benjamin. "Généralisation de l'homologie de Heegaard-Floer aux entrelacs singuliers & raffinement de l'homologie de Khovanov aux entrelacs restreints." Toulouse 3, 2007. http://thesesups.ups-tlse.fr/145/.
Full textAbstract:
La catégorification d'un invariant polynomial d'entrelacs I est un invariant de type homologique dont la caractéristique d'Euler gradue est égale à I. On pourra citer la catégorification originelle du polynôme de Jones par M. Khovanov ou celle du polynôme d'Alexander par P. Ozsvath et Z. Szabo. Outre leur capacité accrue à distinguer les noeuds, ces nouveaux invariants de type homologique semblent drainer beaucoup d'informations d'ordre géométrique. D'autre part, suite aux travaux de I. Vassiliev dans les années 90, un invariant polynomial d'entrelacs peut être étudié à l'aune de certaines propriétés, dites de type fini, de son extension naturelle aux entrelacs singuliers, c'est-à-dire aux entrelacs possédant un nombre fini de points doubles transverses. La première partie de cette thèse s'intéresse aux liens éventuels entre ces deux procédés, dans le cas particulier du polynôme d'Alexander. Dans cette optique, nous donnons d'abord une description des entrelacs singuliers par diagrammes en grilles. Nous l'utilisons ensuite pour généraliser l'homologie de Ozsvath et Szabo aux entrelacs singuliers. Outre la cohérence de sa définition, nous montrons que cet invariant devient acyclique sous certaines conditions annulant naturellement sa caractéristique d'Euler. Ce travail s'insère dans un programme plus vaste de catégorification des théories de Vassiliev. Dans une seconde partie, nous nous proposons de raffiner l'homologie de Khovanov aux entrelacs restreints. Ces derniers correspondent aux diagrammes d'entrelacs quotientés par un nombre restreint de mouvements de Reidemeister. Les tresses fermées apparaissent notamment comme sous-ensemble de ces entrelacs restreints. Un tel raffinement de l'homologie de Khovanov offre donc un nouvel outil pour une étude plus ciblée des noeuds et de leurs déformationsA categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i. E. Links with a finite number of rigid transverse double points. The first part of this thesis deals with a possible relation between these two approaches in the case of the Alexander polynomial. To this purpose, we extend grid presentation for links to singular links. Then we use it to generalize Ozsvath and Szabo invariant to singular links. Besides the consistency of its definition, we prove that this invariant is acyclic under some conditions which naturally make its Euler characteristic vanish. This work can be considered as a first step toward a categorification of Vassiliev theory. In a second part, we give a refinement of Khovanov homology to restricted links. .
8
Zhang, Melissa. "Localization for Khovanov homologies:." Thesis, Boston College, 2019. http://hdl.handle.net/2345/bc-ir:108470.
Full textAbstract:
Thesis advisor: Julia Elisenda GrigsbyThesis 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
9
Söderberg, Christoffer. "Khovanov Homology of Knots." Thesis, Uppsala universitet, Algebra och geometri, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-325842.
Full text
10
McDougall, Adam Corey. "Relating Khovanov homology to a diagramless homology." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/709.
Full textAbstract:
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.
11
Jacobsson, Magnus. "Khovanov homology and link cobordisms /." Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3765.
Full text
12
Boerner, Jeffrey Thomas Conley. "Khovanov homology in thickened surfaces." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/464.
Full textAbstract:
Mikhail Khovanov developed a bi-graded homology theory for links in the 3-sphere. Khovanov's theory came from a Topological quantum field theory (TQFT) and as such has a geometric interpretation, explored by Dror Bar-Natan. Marta Asaeda, Jozef Przytycki and Adam Sikora extended Khovanov's theory to I-bundles using decorated diagrams. Their theory did not suggest an obvious geometric version since it was not associated to a TQFT. We develop a geometric version of Asaeda, Przytycki and Sikora's theory for links in thickened surfaces. This version leads to two other distinct theories that we also explore.
13
Banfield, Ian Matthew. "Baskets, Staircases and Sutured Khovanov Homology." Thesis, Boston College, 2017. http://hdl.handle.net/2345/bc-ir:108149.
Full textAbstract:
Thesis advisor: Julia E. GrigsbyWe 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
14
Clark, David Alan. "Functoriality for the su(3) Khovanov Homology." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2008. http://wwwlib.umi.com/cr/ucsd/fullcit?p3304464.
Full textAbstract:
Thesis (Ph. D.)--University of California, San Diego, 2008.Title from first page of PDF file (viewed June 20, 2008). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 75-76).
15
Salazar-Torres, Dido Uvaldo. "The Khovanov homology of the jumping jack." Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1745.
Full textAbstract:
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.
16
Lee, Ik Jae. "A new generalization of the Khovanov homology." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14170.
Full textAbstract:
Doctor of PhilosophyDepartment of Mathematics
David Yetter
In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.
17
Becker, Hanno [Verfasser]. "Homotopy-Theoretic Studies of Khovanov-Rozansky Homology / Hanno Becker." Bonn : Universitäts- und Landesbibliothek Bonn, 2015. http://d-nb.info/1079273476/34.
Full text
18
Waldron, Jack Willow. "An invariant of link cobordisms from symplectic Khovanov homology." Thesis, University of Cambridge, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.608808.
Full text
19
Saltz, Adam. "The Spectral Sequence from Khovanov Homology to Heegaard Floer Homology and Transverse Links." Thesis, Boston College, 2016. http://hdl.handle.net/2345/bc-ir:106790.
Full textAbstract:
Thesis advisor: John A. BaldwinKhovanov 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
20
Tram, Heather. "Khovanov Homology as an Generalization of the Jones Polynomial in Kauffman Terms." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1987.
Full textAbstract:
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.
21
Hubbard, Diana D. "Properties and applications of the annular filtration on Khovanov homology." Thesis, Boston College, 2016. http://hdl.handle.net/2345/bc-ir:106791.
Full textAbstract:
Thesis advisor: Julia E. GrigsbyThe 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
22
Lee, Eun Soo 1975. "A new structure on Khovanov's homology." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/16904.
Full textAbstract:
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003.Includes bibliographical references (p. 49).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
The purpose of this thesis is proving conjectures in [1] on the Khovanov invariant. Khovanov invariant [6] is an invariant of (relatively) oriented links which is a cohomology theory over the cube of the resolutions of a link diagram. Khovanov invariant specializes to the Jones polynomial by taking graded Euler characteristic. Bar-Natan [1] [2] computed this invariant for the prime knots of up to 11 crossings. From the data, Bar-Natan, Garoufalidis, and Khovanov formulated two conjectures on the value of the Khovanov invariant of an alternating knot [1][4]. We prove those conjectures by constructing a new map on Khovanov's chain complex which, with the original coboundary map, gives rise to a double complex structure on the chain complex.
by Eun Soo Lee.
Ph.D.
23
Kindred, Thomas. "Checkerboard plumbings." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6160.
Full textAbstract:
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.
24
Watson, Liam. "Involutions sur les variétés de dimension trois et homologie de Khovanov." Thèse, 2009. http://www.archipel.uqam.ca/2310/1/D1817.pdf.
Full textAbstract:
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.
25
Chiun-Ming, Liao. "Some computations of Khovanov Homology." 2005. http://www.cetd.com.tw/ec/thesisdetail.aspx?etdun=U0001-0607200510481500.
Full text
26
Liao, Chiun-Ming, and 廖俊旻. "Some computations of Khovanov Homology." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/09435218788520598999.
Full textAbstract:
碩士國立臺灣大學
數學研究所
93
A new link invariant found by Khovanov is a mysterious invariant. The brief idea is to build a chain complex for a knot so that its Euler characteristic is its Jones polynomial, and we can compute the Khovanov homology for this chain complex. It is a link invariant, but the meaning of the terms in it is not yet varified. Instead some masters drill out many properties inside this invariant. One amazing property of the Khovanov homology of prime alternative knots is stated in Dror Bar-Natan''s paper and is proved by Eun Soo Lee. It says that the Khovanov homology of prime alternative knots appears only in two skew parallel lines if we draw them in a table. In this article I want to find some relationship between connected sum and disjoint union of two knots. In Knovanov''s paper he introduce a nice relation between connected sum and disjoint union of two knots. It is long exact sequences, and my computation is relied on it. The program released by Dror Bar-Natan really does great help to me.
27
Rezazadegan, Reza. "Pseudoholomorphic quilts and Khovanov homology." 2009. http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000051896.
Full text
28
Cheng, Zhechi. "Jones grading from symplectic Khovanov homology." Thesis, 2020. https://doi.org/10.7916/d8-x9qh-w954.
Full textAbstract:
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.
29
Bloom, Jonathan Michael. "Monopole Floer homology, link surgery, and odd Khovanov homology." Thesis, 2011. https://doi.org/10.7916/D8000827.
Full textAbstract:
We construct a link surgery spectral sequence for all versions of monopole Floer homology with mod 2 coefficients, generalizing the exact triangle. The spectral sequence begins with the monopole Floer homology of a hypercube of surgeries on a 3-manifold Y, and converges to the monopole Floer homology of Y itself. This allows one to realize the latter group as the homology of a complex over a combinatorial set of generators. Our construction relates the topology of link surgeries to the combinatorics of graph associahedra, leading to new inductive realizations of the latter. As an application, given a link L in the 3-sphere, we prove that the monopole Floer homology of the branched double-cover arises via a filtered perturbation of the differential on the reduced Khovanov complex of a diagram of L. The associated spectral sequence carries a filtration grading, as well as a mod 2 grading which interpolates between the delta grading on Khovanov homology and the mod 2 grading on Floer homology. Furthermore, the bigraded isomorphism class of the higher pages depends only on the Conway-mutation equivalence class of L. We constrain the existence of an integer bigrading by considering versions of the spectral sequence with non-trivial U action, and determine all monopole Floer groups of branched double-covers of links with thin Khovanov homology. Motivated by this perspective, we show that odd Khovanov homology with integer coefficients is mutation invariant. The proof uses only elementary algebraic topology and leads to a new formula for link signature that is well-adapted to Khovanov homology.
30
Putyra, Krzysztof. "On a triply-graded generalization of Khovanov homology." Thesis, 2014. https://doi.org/10.7916/D86971RX.
Full textAbstract:
In this thesis we study a certain generalization of Khovanov homology that unifies both the original theory due to M. Khovanov, referred to as the even Khovanov homology, and the odd Khovanov homology introduced by P. Ozsv´ath, Z. Szab´o, and J. Rasmussen.
The generalized Khovanov complex is a variant of the formal Khovanov bracket introduced by Bar Natan, constructed in a certain 2-categorical extension of cobordisms, in which the disjoint union is a cubical 2-functor, but not a strict one. This allows us to twist the usual relations between cobordisms with signs or, more generally, other invertible scalars. We prove the homotopy type of the complex is a link invariant, and we show how both even and odd Khovanov homology can be recovered. Then we analyze other link homology theories arising from this construction such as a unified theory over the ring Z_p :=Z[p]/(p²−1), and a variant of the algebra of dotted cobordisms, defined over k := Z[X,Y,Z^±1]/(X² = Y² = 1).
The generalized chain complex is bigraded, but the new grading does not make it a stronger invariant. However, it controls up to some extend signs in the complex, the property we use to prove several properties of the generalized Khovanov complex such as multiplicativity with respect to disjoint unions and connected sums of links, and the duality between complexes for a link and its mirror image. In particular, it follows the odd Khovanov homology of anticheiral links is self-dual. Finally, we explore Bockstein-type homological operations, proving the unified theory is a finer invariant than the even and odd Khovanov homology taken together.
31
Kim, Juhyun. "Annular Links with sl₂-Irreducible Annular Khovanov Homology." Thesis, 2021. https://thesis.library.caltech.edu/14149/4/JuhyunKim-2021a.pdf.
Full textAbstract:
We prove that the rank of annular Khovanov homology of a braid in its next-to-top annular grading is always greater than 1, and as an immediate consequence prove that annular Khovanov homology of an annular link as a representation over the Lie algebra sl₂ is irreducible if and only if the annular link is isotopic to the core of the annulus. We also conjecture an analogue of Fox's trapezoid conjecture in the context of annular Khovanov homology with a computer-assisted supporting evidence.
32
"State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology." Thesis, 2010. http://hdl.handle.net/1911/62003.
Full textAbstract:
We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map phi. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov's thickness criteria. We also exhibit a sequence of prime links related by quasipositive modification whose width is increasing.
33
Rose, David Emile Vatcher. "Categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariant of framed tangles." Diss., 2012. http://hdl.handle.net/10161/5592.
Full textAbstract:
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.
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.
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 .
Dissertation