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Un dels problemes oberts més excitants en topologia quàntica es relaciona amb la categorificació de la teoria complexa de Chern-Simons. En aquest context, Gukov i Manolescu introdueixen una sèrie en dues variables $F_K(x,q)$ com la restricció dels invariants de 3-varietats $\hat{Z}$ a complements de nusos, la qual s'ha demostrat que té propietats sorprenents. En particular, aquesta sèrie es pot pensar com una continuació analítica del celebrat polinomi de Jones de teoria de nusos. Subjacent a aquesta construcció s'hi hagi l'elecció d'un grup de Lie $G=SU(2)$. Una pregunta natural concerneix l'estudi d'aquests invariants per a $SU(N)$ de major rang. Fins ara, han hagut intents de definir-los rigorosament, però una definició matemàtica de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ per a nusos no torals encara manca en l'escenari general. Ens centrem en el grup de Lie $SU(3)$. Per a poder estudiar el comportament de l'invariant de nusos associat $F_K^{SU(3)}$, proporcionem una construcció a partir de $R$ matrius dels invariants de nusos cuántidos de $\sl_3$ per a qualsevol nus i representació irreductible finit-dimensional. El resultat principal d'aquesta tesi dona una construcció explícita de $F_K^{SU(3)}(x,y,q)$, que estén la seva prèvia definició a la família de major grandària nus de trena positiva. En particular, es mostra que $F_K^{LA SEVA(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariant quàntic mitjançant especialitzacions en les variables $x$ e $y$., Uno de los problemas abiertos más excitantes en topología cuántica se relaciona con la categorificación de la teoría compleja de Chern-Simons. En este contexto, Gukov y Manolescu introducen una serie en dos variables $F_K(x,q)$ como la restricción de los invariantes de 3-variedades $\hat{Z}$ a complementos de nudos, la cual se ha demostrado que tiene propiedades sorprendentes. En particular, esta serie se puede pensar como una continuación analítica del celebrado polinomio de Jones de teoría de nudos. Subyacente a esta construcción se haya la elección de un grupo de Lie $G=SU(2)$. Una pregunta natural concierne el estudio de estos invariantes para $SU(N)$ de mayor rango. Hasta ahora, han habido intentos de definirlos rigurosamente, pero una definición matemática de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ para nudos no torales todavía carece en el escenario general. Nos centramos en el grupo de Lie $SU(3)$. Para poder estudiar el comportamiento del invariante de nudos asociado $F_K^{SU(3)}$, proporcionamos una construcción a partir de $R$ matrices de los invariantes de nudos cuántidos de $\sl_3$ para cualquier nudo y representación irreducible finito-dimensional. El resultado principal de esta tesis da una construcción explícita de $F_K^{SU(3)}(x,y,q)$, que extiende su previa definición a la familia de mayor tamaño nudos de trenza positiva. En particular, se muestra que $F_K^{SU(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariante cuántico mediante especializaciones en las variables $x$ e $y$., One of the most exciting open problems in quantum topology concerns the categorification of the complex Chern-Simons theory. In this context, Gukov and Manolescu introduce a two--variable series $F_K(x,q)$ as the restriction of the 3-manifold invariant $\hat{Z}$ to knot complements, which has been shown to exhibit surprising properties. In particular, this series can be thought as an analytical continuation of the celebrated colored Jones polynomial of knot theory. Underlying this construction there is the choice of a Lie group $G=SU(2)$. A natural question turns to the study of this invariants for higher rank $SU(N)$. So far, there have been attempts to define them rigorously, but a mathematical definition of $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ for non-torus knots is still lacking in a general scenario. We focus on the Lie group $SU(3)$. In order to study the behaviour of the associated knot invariant $F_K^{SU(3)}$, we provide a construction via $R$-matrices of the $\mathfrak{sl}_3$ quantum knot invariants for any knot and finite-dimensional irreducible representation. The main result of this thesis gives an explicit construction of $F_K^{SU(3)}(x,y,q)$, which extends its former definition to the bigger family of positive braid knots. In particular, it is shown that $F_K^{SU(3)}(x,y,q)$ recovers the $\mathfrak{sl}_3$ quantum invariants via specializations on the $x$ and $y$ variables., Outgoing

A física teòrica, la Teoria Quàntica de Camps (TQC) és un marc teòric extremadament reeixit que combina la Relativitat Especial amb la Mecànica Quàntica, permetent el diseny de models físics de les partícules subatómiques i quasipartícules que descriuen els aspectes més fundamentals de la matèria amb una precisió increïblement alta. Entre aquestes teories, la TQC Chern-Simons és una especial, que no només descriu fenòmens topològics a la física com ara l’Efecte Hall Quàntic, sinó que encaixa amb la noció del que es coneix com una Teoria Topològica de Camps Quàntics (TTCQ). Va ser fent servir aquests axiomes de les TTCQs que Edward Witten va mostrar al 1989 com d’estretament relacionada està la teoria de Chern-Simons amb l’àmbit d’invariants polinòmics que apareixen a la Teoria de Nusos, com ara el ben conegut polinomi de Jones. En aquests darrers anys, investigacions en aquest camp han donat lloc a nous i més poderosos invariants d’enllaços i, a través de cirurgies de Dehn sobre ells, de 3-varietats també. Per exemple, la sèrie de Gukov-Manolescu recentment proposta el 2020 —denotada FK(x, q)— és un invariant conjectural de complements de nusos que, en cert sentit, continua analíticament els polinomis de Jones colorejats. Poc després, Sunghyuk Park va introduir l’enfoc de la Matriu R de Gran Color corresponent a sl(2,C) per estudiar FK per trenats positius i calcular FK per a diversos nusos i enllaços. Aquest procediment ha estat així mateix extès per Angus Gruen a totes les altres àlgebres de Lie sl(n+1) més enllà de sl(2). En aquest treball, després d’un extens repàs sobre els anteriorment esmentats conceptes, abordem la família so(2n) d’àlgebres de Lie semisimples sobre els complexos a la classificació de Cartan, centrant-nos principalment en el cas so(8) atrets per la simetria triple al seu diagrama de Dynkin D4., En física teórica, la Teoría Cuántica de Campos (TCC) es un marco teórico extremadamente exitoso que combina la Relatividad Especial con la Mecánica Cuántica, permitiendo el diseño de modelos físicos de las partículas subatómicas y cuasipartículas que describen los aspectos más fundamentales de la materia con una precisión increíblemente alta. Entre dichas teorías, la TCC Chern-Simons es una especial, que no sólo describe fenómenos topológicos en física tales como el Efecto Hall Cuántico, sino que encaja con la noción de lo que se conoce como una Teoría Topológica de Campos Cuánticos (TTCC). Fue utilizando estos axiomas de las TTCCs que Edward Witten mostró en 1989 cómo de estrechamente relacionada está la teoría de Chern-Simons con el ámbito de invariantes polinómicos que aparecen en la Teoría de Nudos, tales como el bien conocido polinomio de Jones. En estos últimos años, investigaciones en este campo han dado lugar a nuevos y más poderosos invariantes de enlaces y, a través de cirugías de Dehn sobre ellos, así mismo de 3-variedades. Por ejemplo, la serie de Gukov-Manolescu recientemente propuesta en 2020 —denotada FK(x, q)— es un invariante conjetural de complementos de nudos que, en cierto sentido, continúa analíticamente los polinomios de Jones coloreados. Poco después, Sunghyuk Park introdujo el enfoque de la Matriz R de Gran Color correspondiente a sl(2,C) para estudiar FK para trenzados positivos y calcular FK para varios nudos y enlaces. Este procedimiento ha sido a su vez extendido por Angus Gruen a todas las otras álgebras de Lie sl(n+1) más allá de sl(2). En la presente obra, tras un extenso repaso sobre los anteriormente mencionados conceptos, abordamos la família so(2n) de álgebras de Lie semisimples sobre los complejos en la clasificación de Cartan, centrándonos principalmente en el caso so(8) atraídos por la simetría triple en su diagrama de Dynkin D4., In theoretical physics, Quantum Field Theory (QFT) is an extremely successful theoretical framework combining both Special Relativity and Quantum Mechanics, enabling to design physical models of subatomic particles and quasiparticles describing the most fundamental aspects of matter with an incredibly high accuracy. Among these theories, the Chern-Simons QFT is a special one, not only describing topological phenomena in physics such as the Quantum Hall Effect, but also fitting the notion of what is known as a Topological Quantum Field Theory (TQFT). It was by using the axioms of TQFTs that Edward Witten showed back in 1989 how closely related the Chern-Simons theory is to the realm of polynomial invariants appearing in Knot Theory, such as the well-known Jones polynomial. In the past years, further research in this field has led to new and more powerful invariants of links and, by means of Dehn surgeries on them, of 3-manifolds as well. For instance, the Gukov-Manolescu series proposed recently in 2020 —denoted FK(x, q)— is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. Shortly after, Sunghyuk Park introduced the Large Color R-matrix approach for sl(2,C) to study FK for some simple links, giving a definition of FK for positive braid knots and computing FK for various knots and links. This procedure has in turn been extended by Angus Gruen to all other Lie algebras sl(n+1) beyond sl(2). In this work, after a broad review on the above mentioned background, we move on to the family so(2n) of complex semisimple Lie algebras in Cartan’s classification, mainly focusing on the so(8) case attracted by the three-fold symmetry in its Dynkin diagram D4., Outgoing

We study the topological A-twist of 3d N = 4 Yang-Mills gauge theories, with an eye towards geometric representation theory and knot theory. We present an explicit, geometric category describing 1/2-BPS vortex line operators in these theories, as well as collisions of local operators bound to them in terms of convolution techniques generalizing the work of Braverman-Finkelberg-Nakajima on Coulomb branches of vacua. Given a suitable Dirichlet boundary, we show that local operators bound to these vortex line operators can be represented as linear operators between the Borel-Moore homologies of generalized affine Springer fibers, vastly generalizing classical work on affine Springer representations of Hecke algebras and affine Weyl groups. We end with an application to knot theory. We apply 3d mirror symmetry to a recent construction in B-twisted 3d N = 4 gauge theory of HOMFLY-PT knot homology due to Oblomkov-Rozansky to obtain a mirror construction in the A-twist. The mirror construction exactly reproduces a different realization of HOMFLY-PT homology for positive algebraic links due to Oblomkov-Rasmussen-Shende, providing a robust check of our proposed mirror construction.

This international academic symposium focuses on recent progress in mathematical visualization, and will promote discussions between researchers in this field from abroad and in Japan, aiming to encourage future collaborations.

Computer algebra is applied to biomedical computer vision. Specifically certain biomedical images resulting from a case of rhinocerebral mucormysocis in a diabetic boy are analyzed using techniques in computational geometry and in algebraic-geometric topology. We apply convolution and deblurring via diffusion equation from the side of computational geometry and knot theory, graph theory and singular homology form the side of algebraic-geometric topology. Our strategy consists in to represent the biomedical images using algebraic structures in such way that the peculiarities of the images are represented using algebraic complexities. With our strategy we obtain an automatic procedure for the analysis and the diagnostic based on biomedical images. © 2010 Springer-Verlag Berlin Heidelberg.

The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested and illustrated by considering a case study discussed in the literature. On the view put forward here one can obtain a manner of informal, rigorous mathematics founded on any of a variety of proof systems. The latter part of the thesis is concerned with the question of how we should decide which of these competing proof systems we should base our mathematics on: i.e., the question of which proof system we should take as a foundation for our mathematics. A novel answer to this question is proposed, in which the key property we should require of a proof system is that for as many different kinds of structures as possible, when the proof system allows a generalization about that kind of structure to be proved, the generalization actually holds of all real examples of that kind of structure which exist. This is the requirement of soundness of the proof system (for each kind of structure). It is argued that the best way to establish the soundness of a proof system may be by giving an interpretation of its axioms on which they are established as true. As preparation for this discussion, the thesis first investigates the logical and conceptual basis of various arithmetic concepts, with the results obtained used in the final discussion of soundness.

Kauffman monoids Kn and Jones monoids Jn, n=2,3,…, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids K3 and K4 satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in K4. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid J4. © Springer Nature Switzerland AG 2020.

There are many models of physical systems based in unsigned graph theory, however there are some systems that cannot be modelled by unsigned graphs and that is the reason of signed graph states models. For these models, there are useful objects as partition functions, and systems that share these objects have some intrinsic properties in common, so it is interesting to build these models in order to relate those systems (with vertex switching invariance or equivalent knot invariance)., Outgoing

Many well known representations of the braid groups are parameterized by a complex parameter, such as the Burau, Jones and BMW representations. This dissertation develops a construction for choosing specializations of the parameters so the images of the representations are discrete groups. This construction requires not only a parameterized representation, but the representations need to be sesquilinear. Squier showed that the Burau representation is sesquilinear. This dissertation extends Squier's result to all of the Jones and BMW representations, and finds discrete specializations of these representations.

This thesis adapts the cubical CAT(0) Aitchison complex, A(L), of alternating link exteriors to construct CAT(0) polyhedral metric structures, A′(L), on the exterior of various links from cubes and prisms. The first class of links for which this is done is a subclass of adequate links, called suitable adequate links. This class is characterised by the fact that it decomposes into strongly alternating tangles, that satisfy certain mild conditions. An example of a suitable adequate link, used throughout the thesis to illustrate the theory, is 1533593. The positive and negative state sum surfaces of adequate links are essential. Put another way, the checkerboard surfaces of the projection of an adequate link onto its Turaev surface are essential. These two surfaces provide a topological way of characterising an alternating link and they form totally geodesic surfaces in the Aitchison complex. In the suitable adequate case, these essential state sum surfaces are shown to be combinatorially generalised pleated. Elaborating on the technique for constructing A′(L) for suitable adequate links, CAT(0) semi-cubical complexes are constructed for the exterior of links belonging to a class of amenable links, which includes the class of suitable adequate links, but also links that have no adequate diagram, such as 11n07 . Innovations on the boundary pattern of the polyhedral decomposition of these link exteriors, allow the technique to be adapted once more so as a CAT(0) semi-cubing can be placed on the link exterior of adequate Montesinos link (3,−3,3,−2) and the planar 2-cable of the Figure Eight knot. Finally, by increasing the number of pieces in the polyhedral decomposition of the link exterior, a CAT(0) semi- cubing can be placed on the exterior of a class of links dubbed ‘stacked links’: links whose projection onto a higher genus Heegaard surface decompose into strongly alternating tangles, satisfying a variant of the mild condition imposed on the tangles of the class of s

[EN] Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientic visualization, but did not provide a formal proof of self-intersection. An example with a formal existence proof is given, even while the exact self-intersection point remains undetermined.

Persistent homology typically studies the evolution of homology groups Hp(X) (with coefficients in a field) along a filtration of topological spaces. A8-persistence extends this theory by analysing the evolution of subspaces such as V:=Ker¿n|Hp(X)¿Hp(X), where {¿m}m=1 denotes a structure of A8-coalgebra on H*(X). In this paper we illustrate how A8-persistence can be useful beyond persistent homology by discussing the topological meaning of V, which is the most basic form of A8-persistence group. In addition, we explore how to choose A8-coalgebras along a filtration to make the A8-persistence groups carry more faithful information., Peer Reviewed, Postprint (author's final draft)

The low visibility and specialised languages of mathematical work pose challenges for the ethnographic study of communication in mathematics, but observation-based study can offer a real-world grounding to questions about the nature of its methods. This paper uses theoretical ideas from linguistic pragmatics to examine how mutual understandings of diagrams are achieved in the course of conference presentations. Presenters use shared knowledge to train others to interpret diagrams in the ways favoured by the community of experts, directing an audience’s attention so as to develop a shared understanding of a diagram’s features and possible manipulations. In this way, expectations about the intentions of others and appeals to knowledge about the manipulation of objects play a part in the development and communication of concepts in mathematical discourse.

We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log n) steps, while for the latter T(n) steps are always sufficient and sometimes necessary., Postprint (published version)

Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of the more powerful HOMFLY-PT polynomial remained an open problem. Here we show that computing HOMFLY-PT is fixed-parameter tractable in the treewidth, and we give the first sub-exponential time algorithm to compute it for arbitrary links.

In the early 2000s, Ozsv\'{a}th and Szab\'{o} introduced a collection of invariants for 3--manifolds and 4--manifolds called Heegaard Floer homology. To a 3--manifold they constructed a group, and to a 4--manifold which cobounds two 3--manifolds, they constructed a homomorphism between the manifolds appearing on the ends. Their invariants satisfy many of the axioms of a TQFT as described by Atiyah, however their construction has some additional restrictions which prevent it from fitting into Atiyah's framework. There is a refinement of Heegaard Floer homology for 3--manifolds containing a knot, due to Ozsv\'{a}th and Szab\'{o}, and independently Rasmussen, and a further refinement for 3--manifolds containing links, due to Ozsv\'{a}th and Szab\'{o}. It's a natural question as to whether one can define functorial maps associated to link cobordisms.In this thesis, we describe a package of cobordism maps for Heegaard Floer homology and link Floer homology. The cobordism maps satisfy an appropriate analogy of the axiomatic description of a TQFT formulated by Atiyah. To a ribbon graph cobordism between two based 3--manifolds, we associate a map between the Heegaard Floer homologies of the ends. To a decorated link cobordism, we obtain maps on the link Floer homologies of the ends. The maps associated to decorated link cobordisms reduce to the maps for ribbon graphs, in a natural way. As applications, we describe several formulas for mapping class group actions on the Heegaard Floer and knot Floer groups. We prove a new bound on a concordance invariant $\Upsilon_K(t)$ from knot Floer homology, and also see how the link cobordism maps give straightforward proofs of other bounds on concordance invariants from knot Floer homology. We also explore the interaction of the maps with conjugation actions on Heegaard Floer homology and link Floer homology, giving connected sum formulas for involutive Heegaard Floer homology and involutive knot Floer homology.

Knot theory is the mathematical study of knots. In this thesis we study knots and one of its applications in DNA. Knot theory sits in the mathematical field of topology and naturally this is where the work begins. Topological concepts such as topological spaces, homeomorphisms, and homology are considered. Thereafter knot theory, and in particular, knot theoretical invariants are examined, aiming to provide insights into why it is difficult to answer the question "How can we tell knots appart?". In knot theory invariants such as the bracket polynomial, the Jones polynomial and tricolorability are considered as well as other helpful results like Seifert surfaces. Lastly knot theory is applied to DNA, where it will shed light on how certain enzymes interact with the genome.

The topic of study of this thesis belongs both to knot theory and to group theory. A knot is a smooth embedding of a circle in $\mathbb{R}^3$ or $S^3=\mathbb{R}^3\cup\{+\infty\}$. With any knot $K$ we can do an operation which depends on two integer coefficients $p$ and $q$, called $\frac{p}{q}$ Dehn surgery, resulting in a 3-manifold $M$ denoted by $M:=S^3(K,\frac{p}{q})$. A group is left-orderable if it can be given a total strict ordering which is invariant under multiplication from the left. It is hard to understand Dehn surgery geometrically, but algebraically it is clear - the fundamental group $\pi_1(M)$ equals to the fundamental group of the knot complement of $K$ with one relation added. Although the fundamental group of the knot complement is always left-orderable, $\pi_1(M)$ may not be left-orderable. We study left-orderability of $\pi_1(M)$ in case where $K$ is a twisted torus knot.

This report documents the program and the outcomes of Dagstuhl Seminar 17072 "Applications of Topology to the Analysis of 1-Dimensional Objects".

In this paper, we define and study flexible links and flexible isotopy in RP3 subset of CP3. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholm's interpretation of Viro's encomplexed writhe.

Kelly (1990) and Wada (1992) independently identi ed and de ned the generalised knot groups (Gn). The square (SK) and granny (GK) knots are two of the most well-known distinct knots with isomorphic knot groups. Tuffley (2007) confirmed Lin and Nelson's (2006) conjecture that Gn(SK) and Gn(GK) were non-isomorphic by showing that they have different numbers of homomorphisms to suitably chosen finite groups. He concluded that more information about K is carried by generalised knot groups than by fundamental knot groups. Soon after, Nelson and Neumann (2008) showed that the 2-generalised knot group distinguishes knots up to reflection. The goal of this study is to show that for certain square and granny knot analogues, the difference can be detected by counting homomorphisms into a suitable finite groups. This study extends Tuffley's work to analogues SKab and GKab of the square and granny knots formed from connect sums of (a, b)-torus knots. It gives further information about the generalised knot groups of the connect sum of two torus knots, which differ only in their orientation.

We describe a geometrical property of helical structures and show how it accounts for the early art of rope-making. Helices have a maximum number of rotations that can be added to them — and it is shown that this is a geometrical feature, not a material property. This geometrical insight explains why nearly identically appearing ropes can be made from very different materials and it is also the reason behind the unyielding nature of ropes. Maximally rotated strands behave as zero-twist structures. Hence, under strain they neither rotate in one direction nor in the other. The necessity for the rope to be stretched while being laid, known from Egyptian tomb scenes, follows straightforwardly, as does the function of the top, an old tool for laying ropes.

We investigate several natural invariants of curves and knots in $${\mathbb{R}^3}$$ . These invariants generalize bridge number and width. As with bridge number, there are connections to the total curvature of a curve.

We present different means of classifying protein structure. One is made rigorous by mathematical knot invariants that coincide reasonably well with ordinary graphical fold classification and another classification is by packing analysis. Furthermore when constructing our mathematical fold classifications, we utilize standard neural network methods for predicting protein fold classes from amino acid sequences. We also make an analysis of the redundancy of the structural classifications in relation to function and ligand binding. Finally we advocate the use of combining the measurement of the VA, VCD, Raman, ROA, EA and ECD spectra with the primary sequence as a way to improve both the accuracy and reliability of fold class prediction schemes.

This thesis investigates aspects of field theories and soliton solutions with nontrivial topology. In particular we explore the following effective models: a limited sector of the scalar Electroweak theory called extended Abelian Higgs model, and a classical mechanics model derived from the low energy SU(2) Yang-Mills theory. The extended Abelian Higgs model applied on two-component plasma of charged particles is studied numerically. We find evidence that the model admits straight twisted line vortices. The result is described by an energy function that acquires a minimum value for a non-trivial twist. In addition to the twisted line vortices the result also suggests that stable torus shaped solitons are solutions of the theory. Furthermore we construct a classical mechanics model exhibiting some of the key properties of the low-energy Yang-Mills theory. The dynamics of the model is studied numerically. We find that its classical equations of motion support stable periodic orbits. In a three dimensional projection these trajectories are self-linked in a topologically non-trivial manner suggesting the existence of knotted configurations in low energy SU(2) Yang-Mills theory. We calculate the one-loop effective action for the Abelian Higgs model with extended Higgs sector. The resulting first order quantum corrected model shows close resemblance to a modified model where texture stabilizing term has been added to the system. In the limit where the gauge field can be entirely expressed by the scalar fields, the both models become identical suggesting that the theories are closely connected. This implies that quantum corrections have stabilising effect on the soliton solutions. These studies have contributed to a better understanding of the dynamics of non-linear low energy systems, and brought us a step closer to exploring full scale physically realistic models.

Ho Wing Yin., Thesis (M.Phil.)--Chinese University of Hong Kong, 2006., Includes bibliographical references (leaves 67-68)., s in English and Chinese., Chapter 0 --- Introduction --- p.5, Chapter 1 --- Basics of Differential Topology and Knot Theory --- p.7, Chapter 1.1 --- Elementary Differential Topology --- p.7, Chapter 1.2 --- Elementary Knot Theory --- p.13, Chapter 1.2.1 --- Basic Definition --- p.13, Chapter 1.2.2 --- Regular Projection --- p.14, Chapter 1.2.3 --- Classical Knot Invariants --- p.15, Chapter 2 --- Thickness of Knots --- p.17, Chapter 2.1 --- Thickness and its Curvature --- p.17, Chapter 2.2 --- Consequences of the Main Theorem --- p.27, Chapter 2.3 --- Polygonal Representation of Knot --- p.30, Chapter 2.4 --- Self-distance and Main Theorem --- p.35, Chapter 3 --- Energy --- p.37, Chapter 3.1 --- Normal Energy and Symmetric Energy --- p.37, Chapter 3.2 --- Mobius Energy --- p.48, Chapter 3.3 --- Another Formulation of Mobius Energy --- p.57, Chapter 3.4 --- A Non Self-Repulsive Energy Function --- p.62, Bibliography --- p.67, http://library.cuhk.edu.hk/record=b5893010, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Liu Chun-Lung., Thesis (M.Phil.)--Chinese University of Hong Kong, 2006., Includes bibliographical references (leaves 98-100)., Chapter 1 --- Introduction --- p.5, Chapter 2 --- Fundamentals of thickness --- p.7, Chapter 2.1 --- Definition of thickness --- p.9, Chapter 2.2 --- Basic theorem and corollaries --- p.10, Chapter 2.3 --- Equivalent definitions --- p.17, Chapter 3 --- Ropelength minimizer --- p.19, Chapter 3.1 --- Existence of ropelength minimizer --- p.20, Chapter 3.2 --- Some ropelength minimizers --- p.23, Chapter 4 --- Thickness computation --- p.27, Chapter 4.1 --- Definitions --- p.28, Chapter 4.2 --- Octrope algorithm --- p.33, Chapter 4.3 --- Minor improvements --- p.45, Chapter 5 --- Arc presentation --- p.54, Chapter 5.1 --- Definitions --- p.55, Chapter 5.2 --- Basic Theorems --- p.57, Chapter 5.3 --- Ropelength upper bound --- p.64, Chapter 6 --- Hamiltonian knot projection --- p.74, Chapter 6.1 --- Hamiltonian RPG --- p.75, Chapter 6.2 --- Embedding of RPG --- p.78, Chapter 6.3 --- Ropelength upper bound --- p.95, Bibliography --- p.98, http://library.cuhk.edu.hk/record=b5892854, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Hui, Wing San., Thesis submitted in: November 2003., Thesis (M.Phil.)--Chinese University of Hong Kong, 2004., Includes bibliographical references (leaves 58-60)., s in English and Chinese., Chapter 1 --- Introduction --- p.1, Chapter 1.1 --- Introduction --- p.1, Chapter 1.2 --- Outline of Thesis --- p.2, Chapter 2 --- Basic Knowledge of Knot Theory --- p.3, Chapter 2.1 --- Preparation --- p.3, Chapter 2.1.1 --- "Knots, Knot Equivalence and Isotopic Knot" --- p.3, Chapter 2.1.2 --- Tame and Wild Knots --- p.5, Chapter 2.2 --- Some Invariants and Quantities about Knot --- p.7, Chapter 2.2.1 --- Projection of Knot and Crossing Number --- p.7, Chapter 2.2.2 --- Braids and Braid Index --- p.7, Chapter 3 --- Minimal Stick Number --- p.11, Chapter 3.1 --- History and Definition --- p.11, Chapter 3.2 --- Minimal Stick Number on Some Simple Knots --- p.12, Chapter 3.3 --- Some Theorems on the Minimal Stick Number --- p.14, Chapter 4 --- Superbridge Index --- p.22, Chapter 4.1 --- "Definitions of Bridge Index, Superbridge Index and Total Curvature" --- p.22, Chapter 4.2 --- Superbridge Index and Braid Index --- p.25, Chapter 4.3 --- "Relations between Bridge Index, Superbridge Index and Total Curvature" --- p.29, Chapter 4.4 --- Superbridge Index and Minimal Stick Number --- p.36, Chapter 5 --- The Geometric Knot Space --- p.37, Chapter 5.1 --- Definition of the Geometric Knot Space --- p.37, Chapter 5.2 --- "Geometric Equivalence and Topological Properties of the Geometric Knot Space, Geo(n)" --- p.39, Chapter 5.3 --- "The Spaces Geo(3), Geo(4) and Geo(5)" --- p.40, Chapter 5.4 --- Topology of the Space Geo(6) --- p.43, Chapter 6 --- Concluding Remarks --- p.52, Chapter 6.1 --- Other Results on the Minimal Stick Number --- p.52, Chapter 6.2 --- Minimal Stick Number and Superbridge Index of the Torus Knot --- p.54, Chapter 6.3 --- Explorations of the Geometric Knot Spaces --- p.56, Bibliography --- p.58, http://library.cuhk.edu.hk/record=b5892032, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Ku Wah Kwan., Thesis (M.Phil.)--Chinese University of Hong Kong, 2004., Includes bibliographical references (leaves 61-64)., s in English and Chinese., Chapter 1 --- Basic 3-Dimensional Contact Geometry --- p.5, Chapter 1.1 --- Introduction --- p.5, Chapter 1.2 --- Contact Structure --- p.7, Chapter 1.3 --- Darboux's Theorem --- p.11, Chapter 1.4 --- Characteristic Foliation --- p.13, Chapter 1.5 --- More About S3 with The Standard Contact Structure --- p.16, Chapter 2 --- Legendrian Knots --- p.18, Chapter 2.1 --- Basic Definition --- p.18, Chapter 2.2 --- Front Projection --- p.19, Chapter 2.3 --- Classical Legendrian Knot Invariants --- p.22, Chapter 2.3.1 --- Thurston-Bennequin Invariant --- p.22, Chapter 2.3.2 --- Rotation Number --- p.23, Chapter 2.4 --- Stabilization --- p.24, Chapter 3 --- Convex Surface Theory --- p.26, Chapter 3.1 --- Contact Vector Field --- p.26, Chapter 3.2 --- Convex Surfaces --- p.29, Chapter 3.3 --- Flexibility of Characteristic Foliation --- p.34, Chapter 3.4 --- Bennequin Inequality --- p.36, Chapter 3.5 --- Bypass --- p.38, Chapter 3.5.1 --- Modification of Dividing Curves through Bypass --- p.39, Chapter 3.5.2 --- Relation of Bypass and Stabilizing Disk --- p.40, Chapter 3.5.3 --- Finding Bypass --- p.40, Chapter 3.6 --- Tight Contact Structures on Solid Tori --- p.41, Chapter 4 --- Classification Results --- p.42, Chapter 4.1 --- Unknot --- p.43, Chapter 4.2 --- Positive Torus Knot --- p.45, Chapter 5 --- Transverse Knots --- p.50, Chapter 5.1 --- Basic Definition --- p.50, Chapter 5.2 --- Self-linking Number --- p.54, Chapter 5.3 --- Relation between Transverse Knot and Legendrian Knot --- p.55, Chapter 5.4 --- Classification of Unknot and Torus Knot --- p.57, Chapter 6 --- Recent Development --- p.60, Bibliography --- p.61, http://library.cuhk.edu.hk/record=b5892071, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Two knot diagrams describe isotopic knots if and only if they can be connected by a sequence of planar isotopies and Reidemeister moves (cusp-, self-tangency-, and triple point moves). In this dissertation classes of knot diagrams, Reidemeister moves, and relations between sequences of Reidemeister moves are investigated, in a manner inspired by V. Arnold's theory of plane curves. The local knot diagram invariants are classified, and the concept of knot diagram invariants of nite degree is introduced. Invariants of every nite degree, that jump only under triple point moves, are presented. These invariants prove that triple point moves are necessary for connecting some diagrams of isotopic knots. It is shown that there exists no non-trivial knot diagram invariants of nite degree that jump only under self-tangency moves. That is, from the view point of nite degree invariants, the self-tangency move is superfluous. The most refined topological classification of Reidemeister moves is introduced. This classification distinguishes 24 classes of moves. In particular, it distinguishes some triple point moves that only dier in the cyclic ordering in which the three branches appear on the knot. This information is vital when dealing with knot invariants defined through Gauss diagrams. An algorithm to replace any sequence of Reidemeister moves by a sequence of moves of only six out of the 24 classes is given. This gives a useful criterion to determine whether a Gauss diagram function defines a knot invariant. A graphical calculus of diagrammatic knot invariants has been developed by M. Polyak and O. Viro. M. Goussarov proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. In this dissertation the material of Polyak and Viro's note, which contained no proofs, is presented with all proofs and details, in a self-contained form. The refined criterion for a Gauss diagram function to define a knot invariant is used to prove several formulas for

Given a large set of scattered data (xi, yi, fi), a method for selecting a significantly smaller set of knot points which will represent the larger set is described, leading to a package of FORTRAN subroutines. The selection of the knot point locations is based on the minimization of the sum of the squares of the difference between the average number of points per Dirichlet tile and the actual number of points in each tile, subject to the constraint that each knot is located at the centroid of its tile. The pertinent theoretical and computational aspects of the subroutines are introduced and described in detail. Using the least squares thin plate spline approximation method for constructing surfaces, various test surfaces are examined and compared to surfaces obtained using smoothing splines and the bicubic Hermite approximation method. The FORTRAN subroutines are made available to prospective users through a point of contact.

A link of two components (simple closed curves in (S sup 3)) is a boundary link if the components bound disjoint orientable surfaces in (S sup 3). A link is irreducible if the only toroidal surfaces which separate the components are boundaries of regular neighborhoods of the components. In this paper, the author constructs a family of irreducible boundary links with unknotted components. (Author)

The 3-dimensional Poincare conjecture is that a compact, connected, simply connected 3-manifold without boundary is topologically a 3-sphere S sup 3. Despite efforts to prove the conjecture, it has withstood attack. It is known that every orientable 3-manifold may be obtained by removing a collection of disjoint solid tori from S sup 3 and sewing them back differently. In this paper the author examine some of the possibilities for constructing a counterexample to the Poincare conjecture by removing a single solid torus from S sup 3 and sewing it back differently. Actually, they examine not only this process but one analogous to it which they call 'attaching a pillbox to a cube with a knotted hole.' (Author)

The five chapter headings of the Proceedings are titled (1) Triangulation and Classification Problems, (2) Three-Dimensional Manifolds and Knot Theory, (3) Group Actions on Manifolds, (4) Embedding and Unknotting Problems, and (5) Applications of Algebraic and Surgery Manifolds. (Author), Proceedings of the University of Georgia Topology of Manifolds Inst., held at Athens on 11-22 Aug 69. Library of Congress catalog no. 79-111983. Standard Book no. 8410-1018-8.

The purpose of this paper is to present an introduction to the theory of knots and knot groups assuming an intermediate knowledge of group theory and topology on the part of the reader. Thus Chapter I is concerned with knots, equivalent knots, and tame knots. The basic definitions are given to develop the concept of a fundamental group for a topological space. The trivial knot is defined. Chapter II defines knot group and calculates the knot group for the trefoil, figure eight and square knots. The chapter ends with a proof of the existence of nontrivial knots. Chapter III presents two sophisticated methods of calculating knot groups of compound knots. Examples are given to illustrate the process. The development of knots and the fundamental group in Chapter I is based on material from Crowell and Fox [1J. Chapter II is based primarily on material from Crowell and Fox [1] and Fox [2]. The material on tori sequences and 2-sphere sequences in Chapter III is based heavily on a paper by E. E. Posey [4]. The remainder of the chapter, devoted to the van Kampen Theorem is based on material from Crowell and Fox [1]. The books of Hall [3] and Rotman [5] are recommended as references on group theory. For additional information on homotopy theory consult the book by Whitehead [6].

Disertační práce Alternativní ontologie s podtitulem Topologická imaginace a topologický materialismus se zaměřuje na analýzu prostorových fenoménů a prostoru v intencích matematické disciplíny topologie, která se zajímá o?prostory z hlediska teorie množin. Mým cílem je představit topologii jako nástroj nejen pro současnou filosofii, ale i pro uměleckou tvorbu. Pro potřebu disertační práce formuluji dva pojmy: Topologická imaginace a Topologický materialismus. Topologická imaginace je nástroj a metoda, jak tvořit a myslet s vědomím prostoru jakožto dynamické struktury, jež není vázána pouze pevnými geometrickými zákonitostmi. Tato metoda vznikla jako pojmenování mé dlouhodobé umělecké praxe, která je z velké části opřena o?studium prostoru, topologie, teorie uzlů a hledání způsobů jejich aplikace ve vizuálně-umělecké i teoretické práci. Topologický materialismus navrhuji jako pojem, jež spojuje myšlení sítí a nad-dimenzionálních prostorů s filosofickými proudy materialistické tradice, zejména Nového materialismu. Mojí základní tezí je, že tyto směry nelze vnímat odděleně. Materialismus nelze myslet bez jeho prostorové dimenze a topologie bez ukotvení v?materiální světě se stává pouhou abstrakcí. Druhá část disertační práce je věnována analýzám konkrétních prostorů: námi obývaného prostoru, jež nazývám fenomenologickým, infrastruktury, logistického prostoru, informačního prostoru a prostoru kapitálu. Kromě jednotlivých analýz se též zaměřuji na jejich průsečíky, napojení a společné fungování., The dissertation Alternative Ontology, subtitled Topological Imagination and Topological Materialism, focuses on the analysis of spatial phenomena and space in the intentions of the mathematical discipline of topology, which is interested in spaces from the point of view of set theory. My goal is to present topology as a tool not only for contemporary philosophy, but also for artistic creation. For the purpose of the dissertation, I formulate two concepts: Topological imagination and Topological materialism. Topological imagination is a tool and method for creating and thinking with the consciousness of space as a dynamic structure, which is not bound only by fixed laws of geometry. This method originated as the name of my long-term artistic practice, which is largely based on the study of space, topology, knot theory and the search for ways of their application in artistic and theoretical work. I propose Topological materialism as a concept that combines the thinking of networks and multi-dimensional spaces with the philosophical currents of the materialist tradition, especially the New Materialism. My basic thesis is that these cannot be perceived separately. Materialism cannot be thought without its spatial dimension, and topology without anchoring in the material world becomes a mere abstraction. The second part of the dissertation is devoted to the analysis of specific spaces: the space we inhabit, which I call phenomenological, infrastructure, logistics space, information space and the space of capital. In addition to individual analyzes, I also focus on their intersections, connections and joint operation.

Disertační práce Alternativní ontologie s podtitulem Topologická imaginace a topologický materialismus se zaměřuje na analýzu prostorových fenoménů a prostoru v intencích matematické disciplíny topologie, která se zajímá o?prostory z hlediska teorie množin. Mým cílem je představit topologii jako nástroj nejen pro současnou filosofii, ale i pro uměleckou tvorbu. Pro potřebu disertační práce formuluji dva pojmy: Topologická imaginace a Topologický materialismus. Topologická imaginace je nástroj a metoda, jak tvořit a myslet s vědomím prostoru jakožto dynamické struktury, jež není vázána pouze pevnými geometrickými zákonitostmi. Tato metoda vznikla jako pojmenování mé dlouhodobé umělecké praxe, která je z velké části opřena o?studium prostoru, topologie, teorie uzlů a hledání způsobů jejich aplikace ve vizuálně-umělecké i teoretické práci. Topologický materialismus navrhuji jako pojem, jež spojuje myšlení sítí a nad-dimenzionálních prostorů s filosofickými proudy materialistické tradice, zejména Nového materialismu. Mojí základní tezí je, že tyto směry nelze vnímat odděleně. Materialismus nelze myslet bez jeho prostorové dimenze a topologie bez ukotvení v?materiální světě se stává pouhou abstrakcí. Druhá část disertační práce je věnována analýzám konkrétních prostorů: námi obývaného prostoru, jež nazývám fenomenologickým, infrastruktury, logistického prostoru, informačního prostoru a prostoru kapitálu. Kromě jednotlivých analýz se též zaměřuji na jejich průsečíky, napojení a společné fungování., The dissertation Alternative Ontology, subtitled Topological Imagination and Topological Materialism, focuses on the analysis of spatial phenomena and space in the intentions of the mathematical discipline of topology, which is interested in spaces from the point of view of set theory. My goal is to present topology as a tool not only for contemporary philosophy, but also for artistic creation. For the purpose of the dissertation, I formulate two concepts: Topological imagination and Topological materialism. Topological imagination is a tool and method for creating and thinking with the consciousness of space as a dynamic structure, which is not bound only by fixed laws of geometry. This method originated as the name of my long-term artistic practice, which is largely based on the study of space, topology, knot theory and the search for ways of their application in artistic and theoretical work. I propose Topological materialism as a concept that combines the thinking of networks and multi-dimensional spaces with the philosophical currents of the materialist tradition, especially the New Materialism. My basic thesis is that these cannot be perceived separately. Materialism cannot be thought without its spatial dimension, and topology without anchoring in the material world becomes a mere abstraction. The second part of the dissertation is devoted to the analysis of specific spaces: the space we inhabit, which I call phenomenological, infrastructure, logistics space, information space and the space of capital. In addition to individual analyzes, I also focus on their intersections, connections and joint operation.