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The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

Em 1968, J. Milnor introduziu a monodromia local de Picard-Lefschetz de uma hipersuperfície complexa com singularidade isolada. Em seguida, E. Brieskorn perguntou se esta monodromia é sempre finita. Em 1972, Lê Dúng Trâng provou que a resposta é positiva no caso de germes de curvas planas analíticas irredutíveis. Na época, já eram conhecidos exemplos de curvas planas com dois ramos e monodromia finita. Em 1973, N. A?Campo produziu o primeiro exemplo de germe de curva plana com dois ramos e monodromia infinita. Portanto, a questão mais simples, e ainda em aberto, que se coloca neste contexto, é a determinação da finitude da monodromia para germes de curvas planas com dois ramos. O presente trabalho, consiste em determinar, em várias situações, o polinômio mínimo da monodromia de germes de curvas analíticas planas com dois ramos, cujos gêneros são menores ou iguais a dois, o que permite decidir a sua finitude / In 1968, J. Milnor introduced the Picard-Lefschetz monodromy of a complex hypersurface with an isolated singularity. Subsequently, E. Brieskorn asked if this monodromy is always finite. In 1972, Lê Dúng Trâng proved that the answer is positive in the case of irreducible analytic germs of plane curves. At this time, examples of plane curves with two branches and finite monodromy were known. In 1973, N. A?Campo produced the first example of a germ of plane curve with two branches and infinite monodromy. Therefore, the simplest and still open problem in this context is to determine whether the monodromy of a plane curve with two branches is finite or infinite. The present work consists in determining, in several situations, the minimal polynomial of the monodromy for germs of plane analytic curves with two branches, whose genera are less or equal than two, wich allows us to decide its finiteness

Em 1968, J. Milnor introduziu a monodromia local de Picard-Lefschetz de uma hipersuperfície complexa com singularidade isolada. Em seguida, E. Brieskorn perguntou se esta monodromia é sempre finita. Em 1972, Lê Dúng Trâng provou que a resposta é positiva no caso de germes de curvas planas analíticas irredutíveis. Na época, já eram conhecidos exemplos de curvas planas com dois ramos e monodromia finita. Em 1973, N. A?Campo produziu o primeiro exemplo de germe de curva plana com dois ramos e monodromia infinita. Portanto, a questão mais simples, e ainda em aberto, que se coloca neste contexto, é a determinação da finitude da monodromia para germes de curvas planas com dois ramos. O presente trabalho, consiste em determinar, em várias situações, o polinômio mínimo da monodromia de germes de curvas analíticas planas com dois ramos, cujos gêneros são menores ou iguais a dois, o que permite decidir a sua finitude / In 1968, J. Milnor introduced the Picard-Lefschetz monodromy of a complex hypersurface with an isolated singularity. Subsequently, E. Brieskorn asked if this monodromy is always finite. In 1972, Lê Dúng Trâng proved that the answer is positive in the case of irreducible analytic germs of plane curves. At this time, examples of plane curves with two branches and finite monodromy were known. In 1973, N. A?Campo produced the first example of a germ of plane curve with two branches and infinite monodromy. Therefore, the simplest and still open problem in this context is to determine whether the monodromy of a plane curve with two branches is finite or infinite. The present work consists in determining, in several situations, the minimal polynomial of the monodromy for germs of plane analytic curves with two branches, whose genera are less or equal than two, wich allows us to decide its finiteness

In this paper we explore the slice-ribbon conjecture for some families of pretzel knots. Donaldson's diagonalization theorem provides a powerful obstruction to sliceness via the union of the double branched cover W of B⁴ over a slicing disk and a plumbing manifold P([capital gamma]). Donaldson's theorem classifies all slice 4-strand pretzel knots up to mutation. The correction term is another 3-manifold invariant defined by Ozsváth and Szabó. For a slice knot K the number of vanishing correction terms of Y[subscript K] is at least the square root of the order of H₁(Y[subscript K];Z). Donaldson's theorem and the correction term argument together give a strong condition for 5-strand pretzel knots to be slice. However, neither Donaldson's theorem nor the correction terms can distinguish 4-strand and 5-strand slice pretzel knots from their mutants. A version of the twisted Alexander polynomial proposed by Paul Kirk and Charles Livingston provides a feasible way to distinguish those 5-strand slice pretzel knots and their mutants; however the twisted Alexander polynomial fails on 4-strand slice pretzel knots. / text

Cette thèse se situe à l’interface entre théorie des nœuds et théorie des systèmes dynamiques. Le thème central consiste, étant donné un champ de vecteurs dans une variété de dimension 3, à considérer ses orbites périodiques, et à s’interroger sur les informations qu’elles donnent sur le champ de vecteurs et la variété initiaux.La première partie est consacrée au flot géodésique défini sur le fibré unitaire tangentd’une surface, ou d’une orbiface, à courbure constante. L’observation de certains exemples (sphère, tore, surface modulaire) suggère la conjecture suivante, due à Étienne Ghys : l’enlacement entre deux familles homologiquement nulles quelconques d’orbites périodiques est toujours négatif. En d’autres termes, le flot géodésique serait lévogyre. Quand la courbure est négative, par les travaux de David Fried sur les flots d’Anosov, cette conjecture implique une propriété étonnante et très particulière : n’importe quelle collection homologiquement nulle d’orbites périodiques borde une section de Birkhoff pour le flot géodésique, et est par conséquent la reliure d’un livre ouvert. En ce sens, cette conjecture propose une généralisation de la construction de Norbert A’Campo de livres ouverts sur les fibrés unitaires tangents. Nous proposons la démonstration de cette conjecture dans les cas du tore, des orbifolds de type (2, q, infini), et de l’orbifold de type (2, 3, 7). La seconde partie est consacrée au comportement asymptotique des invariants des nœuds formés par les orbites périodiques d’un champ de vecteur, quand la longueur de l’orbite tend vers l’infini. Le but est de définir des invariants de champs de vecteurs stables par difféomorphisme. Dans le cas particulier des nœuds de Lorenz, nous montrons que les racines du polynôme d’Alexander admettent un comportement particulier : elles s’accumulent au voisinage du cercle-unité. / This thesis deals with interactions between knot theory and dynamical systems. Givena vector field on a 3-manifold, the main idea is to study its periodic orbits from the knottheoretical point of view, and to deduce informations about the vector field and the initial manifold. The first part is devoted to the study of the geodesic flow defined on the unit tangent bundle of a surface, or an orbiface, with constant curvature. Simple examples (sphere, torus, modular surface) suggest the following conjecture, due to Ghys : the linking number of two homologically zero collections of periodic orbits is always negative. In other words, the geodesic flow on any orbiface with constant curvature is left-handed. In the negatively curved case, the work of Fried imply another surprising property : any homologically trivial collection of periodic orbits bound a Birkhoff section for the geodesic flow, and is therefore the binding of an open book decomposition. In this setting, the conjecture is a generalization of A’Campo’s construction of open book decompositions on unit tangent bundles. In our work, we prove the conjectre for the torus, for the orbifolds of type (2, q, oo), and for the orbifold of type (2, 3, 7). The second part is devoted to the asymptotic behaviour of invariants of the knots made by the periodic orbits of a vector field, when the length of the orbits tend to infinity. The goal is to define invariants of the vector field under diffeomorphism. In the case of Lorenz knots, we show that the roots of the Alexander polynomial admit an asymptotic behaviour, namely that they accumulate on the unit circle.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21536号 / 理博第4443号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 大槻 知忠, 教授 向井 茂, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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.

The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification of HFL^ is the classical Alexander polynomial for links; likewise, the decategorification of HFT^ gives a local version ∇ˢ of the Alexander polynomial. In the first chapter of this thesis, we give a purely combinatorial definition of this polynomial invariant ∇ˢ via Kauffman states and Alexander codes and investigate some of its properties. As an application, we show that the multivariate Alexander polynomial is mutation invariant. In the second chapter, we define HFT^ in two slightly different, but equivalent ways: One is via Juhász’s sutured Floer homology, the other by imitating the construction of HFL^. We then state a glueing theorem in terms of Zarev’s bordered sutured Floer homology, which endows HFT^ with additional structure. As an application, we show that any two links related by mutation about a (2,−3)-pretzel tangle have the same δ-graded link Floer homology. This result relies on a computer calculation. In the third and last chapter, we specialise to 4-ended tangles. In this case, we give a reformulation of HFT^ with a glueing structure in terms of (what we call) peculiar modules. Together with a glueing theorem, we can easily recover oriented and unoriented skein relations for HFL^. Our peculiar modules also enjoy some symmetry relations, which support a conjecture about δ-graded mutation invariance of HFL^. However, stronger symmetries would be needed to actually prove this conjecture. Finally, we explore the relationship between peculiar modules and twisted complexes in the wrapped Fukaya category of the 4-punctured sphere. There are four appendices, some of which might be of independent interest: In the first appendix, we describe a general construction of dg categories which unifies all algebraic structures used in this thesis, in particular type A and type D modules from bordered theory. In the second appendix, we prove a generalised version of Kauffman’s clock theorem, which plays a major role for our decategorified invariants. The last two appendices are manuals for two Mathematica programs. The first is a tool for computing the generators of HFT^ and the decategorified tangle invariant ∇ˢ. The second allows us to compute bordered sutured Floer homology using nice diagrams.