Дисертації з теми "Transverse knots and links"

Thesis advisor: John A. Baldwin
Contact geometry has played a central role in many recent advances in low-dimensional topology; e.g. in showing that knot Floer homology detects the genus of a knot and whether a knot is fibered. It has also been used to show that the unknot, trefoil, and figure eight knot are determined by their Dehn surgeries. An important problem in 3-dimensional contact geometry is the classification of Legendrian and transverse knots. Such knots come equipped with some classical invariants. New invariants from knot Floer homology have been effective in distinguishing Legendrian and transverse knots with identical classical invariants, a notoriously difficult task. The Giroux correspondence allows contact structures to be studied via purely topological constructs called open book decompositions. Transverse links are then braids about these open books, which in turn may be thought of as mapping tori of diffeomorphisms of compact surfaces with boundary having marked points, which we refer to as pointed monodromies. In the first part of this thesis, we investigate properties of the transverse invariant in knot Floer homology, in particular its behavior for transverse closures of pointed monodromies possessing certain dynamical properties. The binding of an open book sits naturally as a transverse link in the supported contact manifold. We prove that the transverse link invariant in knot Floer homology of the binding union any braid about the open book is non-zero. As an application, we show that any pointed monodromy with fractional Dehn twist coefficient greater than one has non-zero transverse invariant, generalizing a result of Plamenevskaya for braids about the unknot. In the second part of this thesis, we define invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing those defined by Ozsvath, Szabo and Thurston. We show that our invariants are equivalent to those defined by Lisca, Ozsvath, Szabo and Stipsicz for Legendrian and transverse links in arbitrary contact 3-manifolds. Our argument involves considering braids about rational open book decompositions and filtrations on knot Floer complexes
Thesis (PhD) — Boston College, 2019
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

Motivated by the action of XER site-specific recombinase on DNA, this thesis will study the topological properties of a type of local crossing change on oriented knots and links called nullification. One can define a distance between types of knots and links based on the minimum number of nullification moves necessary to change one to the other. Nullification distances form a class of isotopy invariants for oriented knots and links which may help inform potential reaction pathways for enzyme action on DNA. The minimal number of nullification moves to reach a è-component unlink will be called the è-nullification number. This thesis will demonstrate the relationship of the nullification numbers to a variety of knot invariants, and use these to solve the è-nullification numbers for prime knots up to 10 crossings for any è. A table of nullification numbers for oriented prime links up to 9 crossings is also presented, but not all cases are solved. In addition, we examine the families of rational links and torus links for explicit results on nullification. Nullification numbers of torus knots and a subfamily of rational links are solved. In doing so, we obtain an expression for the four genus of said subfamily of rational links, and an expression for the nullity of any torus link.

Knot nullification is an unknotting operation performed on knots and links that can be used to model DNA recombination moves of circular DNA molecules in the laboratory. Thus nullification is a biologically relevant operation that should be studied. Nullification moves can be naturally grouped into two classes: coherent nullification, which preserves the orientation of the knot, and incoherent nullification, which changes the orientation of the knot. We define the coherent (incoherent) nullification number of a knot or link as the minimal number of coherent (incoherent) nullification moves needed to unknot any knot or link. This thesis concentrates on the study of such nullification numbers. In more detail, coherent nullification moves have already been studied at quite some length. This is because the preservation of the previous orientation of the knot, or link, makes the coherent operation easier to study. In particular, a complete solution of coherent nullification numbers has been obtained for the torus knot family, (the solution of the torus link family is still an open question). In this thesis, we concentrate on incoherent nullification numbers, and place an emphasis on calculating the incoherent nullification number for the torus knot and link family. Unfortunately, we were unable to compute the exact incoherent nullification numbers for most torus knots. Instead, our main results are upper and lower bounds on the incoherent nullification number of torus knots and links. In addition we conjecture what the actual incoherent nullification number of a torus knot will be.

This thesis introduces a new quantity called loop number, and shows the conditions in which loop numbers become knot invariants. For a given knot diagram D, one can traverse the knot diagram and count the number of loops created by the traversal. The number of loops recorded depends on the starting point in the diagram D and on the traversal direction. Looking at the minimum or maximum number of loops over all starting points and directions, one can define two positive integers as loop numbers of the diagram D. In this thesis, the conditions under which these loop numbers become knot invariants are identified. In particular, the thesis answers the question when these numbers are invariant under flypes in the diagram D.

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.
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Cataloged from PDF version of thesis.
Includes bibliographical references (pages 65-68).
In this thesis, we use invariants inspired by quantum field theory to study the smooth topology of links in space and surfaces in space-time. In the first half, we use Khovanov homology to the study the relationship between links in R3 and their components. We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the split union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer- Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink. In the second half, we consider knots as potential cross-sections of surfaces in R4. We use Heegaard Floer homology to show that certain knots never occur as cross-sections of surfaces with small first Betti number. (It was previously thought possible that every knot was a cross-section of a connect sum of three Klein bottles.) In particular, we show that any smooth surface in R 4 with cross-section the (2k, 2k - 1) torus knot has first Betti number at least 2k - 2.
by Joshua Batson.
Ph. D.

La tesi propone alcuni esempi di link fibrati in spazi lenticolari. Sfruttando la compatibilità fra le mosse di chirurgia intera e la nozione di open book decomposition, si ricava un esempio di link fibrato prima in L(p,1), per poi generalizzarlo a L(p,q). Si conclude determinando una struttura di contatto equivalente alla open book relativa agli spazi del tipo L(p,1).

Molecular knots and links are intriguing natural phenomena found to spontaneously form in both the biomacromolecules essential for life (e.g. DNA and proteins) and synthetic polymers. As the presence of these entanglements can influence the stability and tensile strength of such molecules, a better understanding of the factors governing their formation and properties is desirable. In this thesis, the synthesis of several new molecular knot and link topologies is described, the majority of which surpass the current scope of interlocked molecules in terms of their structural complexity. The presented strategy utilises the self-assembly of ligand strands and metal cations into circular helicate arrangements, followed by cyclisation of the interwoven complexes through olefin ring-closing metathesis to afford a knot or link. The topological chirality displayed by such molecules is studied, along with their ability to act as receptors for halide anions and metal cations.

Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

Every two-bridge knot or link is characterized by a rational number p/q, and has a fundamental group which has a simple presentation with only two generators and one relator. The relator has a form that gives rise to a formula for the Alexander polynomial of the knot or link in terms of p and q [15]. Every two-bridge knot or link also has a corresponding “up down” graph in terms of p and q. This graph is analyzed combinatorially to prove several properties of the Alexander polynomial. The number of two-bridge knots and links of a given crossing number are also counted.

Thesis advisor: Tao Li
This dissertation explores a relationship between fibered knots and Heegaard splittings in closed, connected, orientable three-manifolds. We show that a fibered knot, which has a sufficiently complicated monodromy, induces a minimal genus Heegaard splitting that is unique up to isotopy. Moreover, we show that fibered knots in the three-sphere has complexity at most 3
Thesis (PhD) — Boston College, 2020
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

In this paper, we examine Fox colorings of virtual knots, and moves called k-swap moves defined for virtual knot diagrams. The k-swap moves induce a one-to-one correspondence between colorings before and after the move, and can be used to reduce the number of virtual crossings. For the study of colorings, we characterize families of alternating virtual knots to generalize (2, n)-torus knots, alternating pretzel knots, and alternating 2-bridge knots. The k-swap moves are then applied to prove a "virtualization" of the Kauffman-Harary conjecture, originally stated for classical knot diagrams, for the above families of virtual pretzel knot diagrams.

The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.

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.

In this dissertation, we study two areas of interest in knot theory: Random knots in the unit cube, and the Yang-Baxter solutions constructed from Frobenius algebras. The study of random knots can be thought of as a model of DNA strings situated in confinement. A random knot with n vertices is a polygonal loop formed by selecting n distinct points in the unit cube, for a positive integer n, and connecting these points by straight line segments successively, such that the last point selected is joined with the first one. We present a step by step description of our algorithm and Maple codes for generating random knots in the unit cube, with a given vertex number n. To detect non-trivial knots, we use a knot invariant called the determinant. We present an algorithm and its Maple code for computing the determinant for random knots. For each vertex number n, we generate large number of random knots and form data sets of values of the determinant. Then we analyze our data sets in various ways. For instance, for each vertex number n, we form data sets of the number of p-colorable random knots by finding the set of prime divisors of each determinant output. We define the stick number for p-colorability to be the minimum number of line segments required to form a p-colorable knot. We use our data sets to find upper bounds for stick numbers for p-colorability, for primes p _ 191. We also find distributions of p-colorable knots and small determinant values. The second topic on random knots is the linking number of random links. A random link is a collection of disjoint random knots produced simultaneously. We present descriptions of our algorithm and its Maple code for constructing random links of two components, and calculating their linking numbers in detail. By running the code for 1000 times, for the vertex number n less than or equal to 30, we obtain data sets of linking numbers for two-component random links such that each component is a random knot with n vertices. Then we find the distribution of linking numbers and calculate upper bounds for the stick number for the linking numbers ` _ 15. The second area we investigate is applications of Fobenius algebras to knot theory. Chain complexes and Yang-Baxter solutions (R-matrices) are constructed by the skein theoretic approach using Frobenius algebras, and deformed R-matrices are constructed by using 2-cocyles. We compute cohomology groups, Yang-Baxter solutions and their cocycle deformations for group algebras, polynomial algebras and complex numbers. We construct knot and link invariants using these R-matrices from Frobenius algebras via Turaev’s criteria. Then a series of skein relations of the invariant are introduced for oriented knot or link diagrams. We also present calculations of the Frobenius skein invariant for various knots and links.

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

We study Seifert surfaces for links, and in particular the Kakimizu complex MS(L) of a link L, which is a simplicial complex that records the structure of the set of taut Seifert surfaces for L. First we study a connection between the reduced Alexander polynomial of a link and the uniqueness of taut Seifert surfaces. Specifically, we reprove and extend a particular case of a result of Juhasz, using very different methods, showing that if a non-split homogeneous link has a reduced Alexander polynomial whose constant term has modulus at most 3 then the link has a unique incompressible Seifert surface. More generally we see that this constant term controls the structure of any non-split homogeneous link. Next we give a complete proof of results stated by Hirasawa and Sakuma, describing explicitly the Kakimizu complex of any non-split, prime, special alternating link. We then calculate the form of the Kakimizu complex of a connected sum of two non-fibred links in terms of the Kakimizu complex of each of the two links. This has previously been done by Kakimizu when one of the two links is fibred. Finally, we address the question of when the Kakimizu complex is locally infinite. We show that if all the taut Seifert surfaces are connected then MS(L) can only be locally infinite when L is a satellite of a torus knot, a cable knot or a connected sum. Additionally we give examples of knots that exhibit this behaviour. We finish by showing that this picture is not complete when disconnected taut Seifert surfaces exist.

Thesis advisor: Joshua E. Greene
We construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 3-braid knot K in S^3, where r < 2g(K)-1 and g(K) denotes the Seifert genus of K. This confirms a prediction of the L--space conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot P(-2,3,7), and indeed along every pretzel knot P(-2,3,q), for q a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. We adapt our techniques to construct taut foliations in every closed 3-manifold obtained along r-framed Dehn surgery along a positive 1-bridge braid, and indeed, along any positive braid knot, in S^3, where r < g(K)-1. These are the only examples of theorems producing taut foliations in surgeries along hyperbolic knots where the interval of surgery slopes is in terms of g(K)
Thesis (PhD) — Boston College, 2020
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

L’A-polinomi d’un nus en S3 és un poliomi de dues variables obtingut projectant la varietat de SL2C-caràcters de l’exterior del nus sobre la varietat de caràcters del grup perifèric. Distingeix el nus trivial i detecta alguns pendents a la vora de superfícies essencials dels exteriors de nus. El concepte de A-polinomi va ser generalitzat a les 3-varietats amb vores tòriques no connexes; una 3-varietat M amb n tors de vora produeix un sub-espai algebraic E(M) de C2n anomenat varietat de valors propis de M. Té dimensió maximal n i E(M) també detecta sistemes de pendents a les vores de superfícies essencials en M. La varietat de valors propis de M sempre conté una part Ered(M), de dimensió maximal, produïda pels caràcters reductibles. Si M és hiperbòlica, E(M) conté una altra component de dimensió maximal; saber quines altres 3-varietats compleixen això encara és una pregunta oberta. En aquesta tesi, estudiem aquest assumpte per dues famílies de 3-varietats amb vores tòriques i, amb dues tècniques diferents, aportem una resposta positiva en ambdós casos. Primerament, estudiem els enllaços Brunnians en S3, enllaços per els quals tot subenllaç estricte és trivial. Algunes propietats d’aquests enllaços i llur estabilitat sota alguns ompliments de Dehn permet mostrar que, si M és l’exterior d’un enllaç Brunnià no trivial i diferent de l’enllaç de Hopf, E(M) conté una component de dimensió maximal diferent de Ered(M). Aquest resultat s’obté generalitzant la tècnica prèviament utilitzada per els nusos en S3 fent servir el teorema de Kronheimer-Mrowka. Per altre banda, considerem una família de varietats-enllaç, varietats obtingudes com exteriors d’enllaços en esferes d’homologia entera. Les varietats-enllaç tenen sistemes perifèrics estàndards de meridans i longituds i són estables per splicing, l’enganxament de dues varietats-enllaç al llarg de tors perifèrics, identificant el meridià de cada costat amb la longitud oposada. El splicing indueix una noció de descomposició tòrica per les varietatsenllaç i anomenem grafejades les varietats-enllaç que admeten una descomposició tòrica per la qual totes les peces són fibrades de Seifert. Mostrem que, excloent els casos trivials, totes les varietats-enllaç grafejades produeixen una altre component de dimensió maximal en les seves varietats de valors propis. Per aquesta segona demostració, presentem una nova generalització de la varietat de valors propis, que també té en compte tors interns, i que presentem en el context més general d’arbres abelians de grups. Un arbre de grup és abelià quan tots els grups de arestes són commutatius; en aquest cas, definim la varietat de valors propis d’un arbre abelià de grup, una varietat algebraica compatible amb dues operacions naturales sobre els arbres: la fusió i la contracció. Això permet estudiar la varietat de valors propis d’una varietat-enllaç mitjançant les varietats de valors propis de les seves descomposicions tòriques. Combinant resultats generals sobre varietats de valors propis d’arbres abelians de grup i les descripcions combinatòries de les varietats-enllaç grafejades, construïm components de dimensió maximal en les seves varietats de valors propis.
Le A-polynôme d’un noeud dans S3 est un polynôme à deux variables obtenu en projetant la variété des SL2C-caractères de l’extérieur du noeud sur la variété de caractères du groupe périphérique. Il distingue le noeud trivial et détecte certaines pentes aux bords de surfaces essentielles des extérieurs de noeud. La notion de A-polynôme a été généralisée aux 3-variétés à bord torique non connexe ; une 3-variétéM bordée par n tores produit un sous-espace algebrique E(M) de C2n appelé variété des valeurs propres deM. Sa dimension est inférieure ou égale à n et E(M) détecte également des systèmes de pentes aux bords de surfaces essentielles dans M. La variété des valeurs propres de M contient toujours un sous-ensemble Ered(M) produit par les caractères réductibles, et de dimension maximale. Si M est hyperbolique, E(M) contient une autre composante de dimension maximale ; pour quelles autres 3- variétes est-ce le cas reste une question ouverte. Dans cette thèse, nous étudions cette question pour deux familles de 3-variétés à bords toriques et, via deux techniques distinctes, apportons une réponse positive dans ces deux cas. Dans un premier temps, nous étudions les entrelacs Brunniens dans S3, entrelacs pour lesquels tout sous-entrelacs strict est trivial. Certaines propriétés de ces entrelacs, et leur stabilité par certains remplissages de Dehn nous permettent de prouver que, siM est l’extérieur d’un entrelacs Brunnien non trivial et différent de l’entrelacs de Hopf, E(M) contient une composante de dimension maximale différente de Ered(M). Ce résultat est obtenu en généralisant la technique préalablement utilisée pour les noeuds dans S3 grâce au théorème de Kronheimer-Mrowka. D’autre part, nous considérons une famille de variétés-entrelacs, variétés obtenues comme extérieurs d’entrelacs dans des sphères d’homologie entière. Les variétés-entrelacs possèdent des systèmes périphériques standard de méridiens et longitudes et sont stables par splicing, le recollement de deux variétés-entrelacs le long de tores périphériques en identifiant le méridien de chaque coté avec la longitude opposée. Ceci induit une notion de décomposition torique de variété-entrelacs et une telle variété est dite graphée si elle admet une décomposition torique où toutes les pièces sont fibrées de Seifert. Nous montrons que, mis-à-part les cas triviaux, toutes les variétés-entrelacs graphées produisent une autre composante de dimension maximale dans leur variétés des valeurs propres. Pour cette seconde preuve, nous présentons une nouvelle généralisation de la variété des valeurs propres, qui prend également en compte les tores intérieurs, que nous introduisons dans le contexte plus général des arbres abéliens de groupes. Un arbre de groupe est appelé abélien si tous les groupes d’arête sont commutatifs ; dans ce cas, nous définissions la variété des valeurs propres d’un arbre abélien de groupe, une variété algébrique compatible avec deux opérations naturelles sur les arbres : la fusion et la contraction. Ceci permet d’étudier la variété des valeurs propres d’une variété-entrelacs à travers les variétés des valeurs propres de ses décompositions toriques. En combinant des résultats généraux sur les variétés des valeurs propres d’arbres abéliens de groupe et les descriptions combinatoires des variétés-entrelacs graphées, nous contruisons des composantes de dimension maximale dans leur variétés des valeur propres.
The A-polynomial of a knot in S3 is a two variable polynomial obtained by projecting the SL2C-character variety of the knot-group to the character variety of its peripheral subgroup. It distinguishes the unknot and detects some boundary slopes of essential surfaces in knot exteriors. The notion of A-polynomial has been generalized to 3-manifolds with non-connected toric boundaries; ifM is a 3-manifold bounded by n tori, this produces an algebraic subset E(M) of C2n called the eigenvalue variety of M. It has dimension at most n and still detects systems of boundary slopes of surfaces in M. The eigenvalue variety of M always contains a part Ered(M) arising from reducible characters and with maximal dimension. If M is hyperbolic, E(M) contains another topdimensional component; for which 3-manifolds is this true remains an open question. In this thesis, this matter is studied for two families of 3-manifolds with toric boundaries and, via two very different technics, we provide a positive answer for both cases. On the one hand, we study Brunnian links in S3, links in the standard 3-sphere for which any strict sublink is trivial. Using special properties of these links and stability under certain Dehn fillings we prove that, if M is the exterior of a Brunnian link different from the trivial link or the Hopf link, then E(M) admits a top-dimensional component different from Ered(M). This is achieved generalizing the technic applied to knots in S3, using Kronheimer-Mrowka theorem. On the other hand, we consider a family of link-manifolds, exteriors of links in integerhomology spheres. Link-manifolds are equipped with standard peripheral systems of meridians and longitudes and are stable under splicing, gluing two link-manifolds along respective boundary components, identifying the meridian of each side to the longitude of the other. This yields a well-defined notion of torus decomposition and a link-manifold is called a graph link-manifold if there exists such a decomposition for which each piece is Seifert-fibred. Discarding trivial cases, we prove that all graph link-manifolds produce another top-dimensional component in their eigenvalue variety. For this second proof, we propose a further generalization of the eigenvalue variety that also takes into account internal tori and this is introduced in the broader context of abelian trees of groups. A tree of group is called abelian if all its edge groups are commutative; in that case, we define the eigenvalue variety of an abelian tree of groups, an algebraic variety compatible with two natural operations on trees: merging and contraction. This enables to study the eigenvalue variety of a link-manifold through the eigenvalue varieties of its torus splittings. Combining general results on eigenvalue varieties of abelian trees of groups with combinatorial descriptions of graph link-manifolds, we construct top-dimensional components in their eigenvalue varieties.

Este trabalho dedica-se à interpretação do conceito de renormalização em sistemas dinâmicos não autónomos periódicos gerados pela iteração sequencial de aplicações do tipo de Lorenz. Para tal socorremo-nos da dinâmica simbólica e do produto estrela sobre os invariantes de amassamento. Começamos por decompor o espaço de fases simbólico de sistemas renormalizáveis e em seguida estudamos a entropia topológica destes sistemas restringidos aos intervalos de renormalização. Finalmente, interpretamos estes conceitos no contexto dos templates com vários segmentos de ramificação, obtendo uma descrição geométrica dos nós e elos correspondentes a órbitas de pontos nos intervalos de renormalização e apresentando fórmulas explícitas para o cálculo do genus destes nós e elos; ABSTRACT: Symbolic dynamics of renormalizable multimodal applications, renormalization in templates This work is dedicated to the interpretation of renormalization of periodic nonautonomous dynamical systems generated by the sequential iteration of Lorenz like applications. For this we use symbolic dynamics and star product on the kneading invariants. We start by decomposing the symbolic phase space of renormalizable systems and then we study the topological entropy of these systems restricted the renormalization intervals. Finally, we interpret these concepts in the context of templates with multiple branching segments, obtaining a geometric description of the knots and links corresponding to orbits of points in renormalization intervals and featuring explicit formulas for calculating the genus of these knots and links.

The aim of this thesis is the study of transverse link invariants coming from Khovanov sl 2 - and sl 3 -homologies and from their deformations. As a by-product of our work we get computable estimates on some concordance invariants coming from Khovanov sl_2-homologies.

This thesis presents an investigation of many known polynomial invariants of knots and links. Following Alexander's original idea, we define another multi-indeterminant polynomial for links and show that it satisfies some of Torres' conditions. We conjecture that they are equivalent.

Conway polynomials have been known since the sixties. In this paper, we show that the polynomials of various orientations of a link are related, at least in the first and second coefficients. The relationship can be expressed as a function of the Conway polynomials of all sublinks.

A new invariant polynomial of knots and links has been discovered which is independent of the orientation. This polynomial is also invariant of link inverses. Moreover, it is different from the Conway polynomial and the newly discovered HOMFLY polynomial. It distinguishes the trivial 3-unlink and the Borremean ring of 3 components. Various properties of the polynomial are studied.

The organization of amino acids in the protein is usually described in terms of four levels of structure classification which, however, misses some important aspects of protein geometry. One of the protein features absent is the existence of the knot tied on the protein backbone. The discovery of such knotted proteins raises the questions of the folding of such proteins and the function of the backbone knot. Despite theoretical and experimental investigation, the answers on both of these questions remain elusive. Moreover, apart from the knotted proteins, some singular cases of other topologically non-trivial proteins were recently identified, for which the folding and the function are also unknown. This work is the first holistic elaboration on the whole field of the proteins with complex topology. Apart from the backbone knots, the work describes also other motifs, some discovered as the result of the project: complex lassos, protein links, knotted loops, and theta-curves. The work concentrates on the classification, occurrence, function, and folding of proteins with the topologically complex motifs. In the classification part, all the topologically non-trivial motifs present in proteins are described. In particular, novel mathematical tools to classify the complex lasso structures are proposed. In the part devoted to occurrence of the motifs, their statistical probability is presented. Observed underrepresentation of the motifs in comparison with polymer models becomes a prelude to the function of the complex topology. In particular, the links are shown to stabilize the structure, and the lasso topology is strongly suggested to be crucial for the function of some proteins. In this part also the enzyme-favoring function of the backbone knot is proposed. The novel, ribosome-based mechanism of folding of the proteins with backbone knots begins the fourth part, in which also the influence of the topology, confinement, and knot tails on folding process is analyzed. The scrupulous analysis of the whole database of the protein structures was possible only with the creation of the special tools. These were given to the broad scientific community in the form of databases, servers, plugins, and a Python package, to which the fifth part of the work is devoted. The work is finalized with the future directions and further reading sections which, hopefully, will inspire younger adepts to immerse into the field of complex topology proteins.
Łańcuchy białkowe opisywane są zazwyczaj w ramach czterorzędowej organizacji struktury. Jednakże, ten sposób opisu nie pozwala na uwzględnienie niektórych aspektów geometrii białek. Jedną z brakujących cech jest obecność węzła stworzonego przez łańcuch główny. Odkrycie białek posiadających taki węzeł budzi pytania o zwijanie takich białek i funkcję węzła. Pomimo połączonego podejścia teoretycznego i eksperymentalnego, odpowiedź na te pytania nadal pozostaje nieuchwytna. Z drugiej strony, prócz zawęźlonych białek, w ostatnich czasach zostały zidentyfikowane pojedyncze struktury zawierające inne, topologicznie nietrywialne motywy. Funkcja tych motywów i ścieżka zwijania białek ich zawierających jest również nieznana w większości przypadków. Ta praca jest pierwszym holistycznym podejściem do całego tematu nietrywialnej topologii w białkach. Prócz białek z zawęźlonym łańcuchem głównym, praca opisuje także inne motywy: białka-lassa, sploty, zawęźlone pętle i teta-krzywe. Niektóre spośród tych motywów zostały odkryte w ramach pracy. Wyniki skoncentrowano na klasyfikacji, występowaniu, funkcji oraz zwijaniu białek z topologicznie nietrywialnymi motywami. W części poświęconej klasyfikacji, zaprezentowane zostały wszystkie topologicznie nietrywialne motywy występujące w białkach. W szczególności, zaproponowano i opisano nowe matematyczne narzędzia umożliwiające klasyfikację białek-lass. W części dotyczącej występowania struktur rozważane jest statystyczne prawdopodobieństwo występowania różnych motywów. Ich mniejsza liczba w porównaniu z szacunkami wynikającymi z modeli polimerowych stanowi wstęp do rozważań na temat funkcji nietrywialnej topologii. W szczególności pokazano, że funkcją splotu jest wprowadzenie szczególnej stabilności łańcucha, a w przypadku niektórych białek topologia lassa jest najprawdopodobniej niezbędna do pełnienia przez nie funkcji. W tej części zaproponowana została również funkcja węzła w łańcuchu głównym, wspomagająca tworzenie i stabilizująca miejsca aktywne enzymów. Nowy mechanizm zwijania zawęźlonych białek wykorzystujący rybosom rozpoczyna część czwartą, w której analizowany jest również wpływ topologii, ograniczonej objętości i długości węzła na zwijanie białek. Skrupulatna analiza wszystkich dostępnych struktur przestrzennych białek możliwa była jedynie po stworzeniu odpowiednich narzędzi programistycznych. Narzędzia te zostały przekazane naukowej wspólnocie pod postacią baz danych, serwerów, wtyczek do innych programów oraz paczki programistycznej. Narzędzia te opisane są w części piątej. Praca kończy się wskazaniem przyszłych kierunków rozwoju dziedziny oraz zbiorem literatury okalającej zagadnienia zawarte w pracy. Zestaw ten skierowany jest do przyszłych adeptów, stanowiąc przewodnik po świecie białek o skomplikowanej topologii i zachętę do dalszych prac.

In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation.

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2019.
Los nudos, tal cual aparecen en nuestra vida cotidiana, son un objeto de estudio en la Matemática. La Teoría de Nudos es la rama de la Matemática que se encarga de su estudio. Un problema central es el de poder decir si dos nudos dados son equivalentes o no. Los matemáticos, en la búsqueda de responder esta pregunta, entre otras, han desarrollado diversas técnicas y herramientas en esta área de estudio. En este trabajo se hace un recorrido en el estudio de la Teoría de Nudos, comenzando con las definiciones más elementales, hasta llegar a estudiar herramientas sofisticadas como el polinomio de Alexander, el grupo de un nudo y las matrices de Seifert, entre otros. En los dos últimos capítulos se investigan los dos temas siguientes: nudos virtuales y presentaciones de Wirtinger. En este último se hace un aporte, dando una nueva familia infinita de presentaciones de Wirtinger no geométricas.
The knots we usually see in our lifes are studied in mathematics in the branch called Knot Theory. A main problem is to decide whether two knots are equivalent or not. Many tools and techniques have been developed by mathematicians in order to answer this and other related questions. In this work, we study Knot Theory from the beginning, with definitions and elementary notions, until some sophisticated concepts and tools like the Alexander polynomial, the knot group and Seifert matrices, among others. In the last two chapters, we work on the following two particular subjects: virtual knots and Wirtinger presentations. In this last one, we made a small contribution by presenting a new infinite family of Wirtinger presentations which are not geometric.
Fil: Gutierrez Quispe. Robert Gerson. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.