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 Title
 3Manifolds of S1Category Three.
 Creator

Wang, Dongxu, Heil, Wolfgang, Niu, Xufeng, Klassen, Eric P., Hironaka, Eriko, Nichols, Warren D., Department of Mathematics, Florida State University
 Abstract/Description

I study 3manifold theory, which is a fascinating research area in topology. Many new ideas and techniques were introduced during these years, which makes it an active and fast developing subject. It is one of the most fruitful branches of today's mathematics and with the solution of the Poincare conjecture, it is getting more attention. This dissertation is motivated by results about categorical properties for 3manifolds. This can be rephrased as the study of 3manifolds which can be...
Show moreI study 3manifold theory, which is a fascinating research area in topology. Many new ideas and techniques were introduced during these years, which makes it an active and fast developing subject. It is one of the most fruitful branches of today's mathematics and with the solution of the Poincare conjecture, it is getting more attention. This dissertation is motivated by results about categorical properties for 3manifolds. This can be rephrased as the study of 3manifolds which can be covered by certain sets satisfying some homotopy properties. A special case is the problem of classifying 3manifolds that can be covered by three simple S1contractible subsets. S1contractible subsets are subsets of a 3manifold M3 that can be deformed into a circle in M3. In this thesis, I consider more geometric subsets with this property, namely subsets are homeomorphic to 3balls, solid tori and solid Klein bottles. The main result is a classication of all closed 3manifolds that can be obtained as a union of three solid Klein bottles.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7650
 Format
 Thesis
 Title
 Character Varieties of Knots and Links with Symmetries.
 Creator

Sparaco, Leona H., Petersen, Kathleen L., Harper, Kristine, Ballas, Sam, Bowers, Philip L., Hironaka, Eriko, Florida State University, College of Arts and Sciences, Department...
Show moreSparaco, Leona H., Petersen, Kathleen L., Harper, Kristine, Ballas, Sam, Bowers, Philip L., Hironaka, Eriko, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of twobridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and...
Show more: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of twobridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and exploit this symmetry to factor the character variety. We then find the geometric genus of both components of the character variety. We compute the SL2(C) character variety for the Borromean ring complement in S^3. Further, we explore how the symmetries effect this character variety. Finally, we prove some general results about the structure of character varieties of links with symmetries.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Sparaco_fsu_0071E_13851
 Format
 Thesis
 Title
 ChernSchwartzMacpherson Classes of Graph Hypersurfaces and Schubert Varieties.
 Creator

Stryker, Judson P., Aluﬃ, Paolo, Van Engelen, Robert, Aldrovandi, Ettore, Hironaka, Eriko, Van Hoeij, Mark, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of...
Show moreThis dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of certain families of these varieties is proven. The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation. In the second chapter, a series of equivalences of graph hypersurfaces that are useful for reducing the number of cases that must be calculated are developed. A table of CSM classes of all but one graph with 6 or fewer edges are explicitly computed. This table also contains Fulton Chern classes and Milnor classes for the graph hypersurfaces. Using the equivalences and a series of formulas from a paper by Aluffi and Mihalcea, a new series of formulas for the CSM classes of certain families of graph hypersurfaces are deduced. I prove positivity for all graph hypersurfaces corresponding to graphs with first Betti number of 3 or less. Formulas for graphs equivalent to graphs with 6 or fewer edges are developed (as well as cones over graphs with 6 or fewer edges). In the third chapter, CSM classes of Schubert varieties are discussed. It is conjectured by Aluffi and Mihalcea that all Chern classes of Schubert varieties are represented by effective cycles. This is proven in special cases by B. Jones. I examine some positivity results by analyzing and applying combinatorial methods to a formula by Aluffi and Mihalcea. Positivity of what could be considered the ``typical' case for low codimensional coefficients is found. Some other general results for positivity of certain coefficients of Schubert varieties are found. This technique establishes positivity for some known cases very quickly, such as the codimension 1 case as described by Jones, as well as establishing positivity for codimension 2 and families of cases that were previously unknown. An unexpected connection between one family of cases and a second order PDE is also found. Positivity is shown for all cases of codimensions 14 and some higher codimensions are discussed. In both the graph hypersurfaces and Schubert varieties, all calculated ChernSchwartzMacPherson classes were found to be positive.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1531
 Format
 Thesis
 Title
 Constructing NonTrivial Elements of the ShafarevichTate Group of an Abelian Variety.
 Creator

Biswas, Saikat, Agashe, Amod, Aggarwal, Sudhir, Hironaka, Eriko, Van Hoeij, Mark, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non...
Show moreThe ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing nontrivial elements of the ShafarevichTate group of an elliptic curve by means of the MordellWeil group of an ambient curve. In this thesis, we extract a general theorem out of Cremona and Mazur's work and give precise conditions under which such a construction can be made. We then give an extension of our result which provides new theoretical evidence for the BSD conjecture. Finally, we prove a theorem that gives an alternative method to potentially construct nontrivial elements of the ShafarevichTate group of an elliptic curve by using the component groups of a second curve.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3717
 Format
 Thesis
 Title
 Dirac Operators, Multipliers and H[superscript P] Spaces of Monogenic Functions.
 Creator

Wang, Guanghou, Nolder, Craig, Hawkes, Lois, Case, Bettye, Hironaka, Eriko, Quine, Jack, Seppälä, Mika, Department of Mathematics, Florida State University
 Abstract/Description

We have done a few things under Clifford algebra settings. Firstly, one Caccioppoli type estimate is derived for solutions of $A$Dirac equations in the form $DA(x,Du) = 0$, where $D$ is the Dirac operator. This kind of $A$Dirac equations are generalizations of elliptic equations of $A$harmonic type, i.e. div$A(x,\nabla u)=0.$ Secondly, the multiplier theory from Fourier analysis is generalized to Clifford analysis. After the multipliers of operators $\mathcal{D}$, $T$ and $ \Pi$ are...
Show moreWe have done a few things under Clifford algebra settings. Firstly, one Caccioppoli type estimate is derived for solutions of $A$Dirac equations in the form $DA(x,Du) = 0$, where $D$ is the Dirac operator. This kind of $A$Dirac equations are generalizations of elliptic equations of $A$harmonic type, i.e. div$A(x,\nabla u)=0.$ Secondly, the multiplier theory from Fourier analysis is generalized to Clifford analysis. After the multipliers of operators $\mathcal{D}$, $T$ and $ \Pi$ are identified, some related properties will be very easy to achieve, including two integral representation theorems, also the iterations of operators $\mathcal{D}$ and $\Delta$ are also discussed. Thirdly, one Carleson measure theorem is achieved for monogenic Hardy spaces on the unit ball in $R^{n+1}$, as well as one Clifford Riesz representation theorem. Furthermore, one bounded theorem about certain inhomogeneous Dirac equations is established with the help of spherical monogenic functions theory.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5259
 Format
 Thesis
 Title
 Finite Abelian Group Actions on Orientable Circle Bundles over Surfaces.
 Creator

Ibrahim, Caroline Maher Boulis, Heil, Wolfgang, Hollander, Myles, Hironaka, Eriko, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

A finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that...
Show moreA finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that preserves the group action. This topological definition is translated to an algebraic definition as two Gactions are equivalent if and only if the associated surjections into G are equivalent via an automorphism of the fundamental group of M. For the manifolds M considered in this dissertation every automorphism of the fundamental group of M can be realized by a homeomorphism of M. Hence there is a onetoone correspondence between the topological and algebraic equivalence. The problem of classifying fixedpoint free finite abelian group actions on surfaces had been investigated by, among others, Nielsen, Smith and Zimmermann. Nielsen classifies cyclic actions on surfaces. He gives a list of automorphisms which he uses in his classification. Smith does the classification for special abelian groups. His approach is different from Nielsen's in the algebraic methods he uses. Zimmermann gives an algebraic solution to the classification of any finite abelian group action on closed surfaces. His technique is to get every surjective homomorphism from the fundamental group of the surface into G in normal form and then differentiate between the normal forms. In this dissertation we classify fixedpoint free finite abelian group actions on circle bundles. By results of Waldhausen every homeomorphism of M is isotopic to a fiber preserving homeomorphism; that is, it preserves the S1 factor of the bundle. This corresponds to the algebraic condition that any automorphism on the fundamental group of M preserves the center of the group. We use the same approach as that of Nielsen on surfaces. We give algorithms to bring every surjective homomorphism from the fundamental group of the bundle into the group G to normal form. From there we differentiate between the normal forms based on Nielsen's results. The results obtained are for circle bundles over surfaces of genus g greater than or equal to 2. A complete classification is given for the case that the circle bundle is a product bundle and G is a finite abelian group. We also obtain a complete classification of cyclic group actions and finite abelian group actions on circle bundles that are not product bundles.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3887
 Format
 Thesis
 Title
 Flow Equivalence Classes of PseudoAnosov Surface Homeomorphisms.
 Creator

Billet, Robert, Hironaka, Eriko, Petersen, Kathleen L., Duke, Dennis, Fenley, Sergio, Heil, Wolfgang, Florida State University, College of Arts and Sciences, Department of...
Show moreBillet, Robert, Hironaka, Eriko, Petersen, Kathleen L., Duke, Dennis, Fenley, Sergio, Heil, Wolfgang, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation explores pseudoAnosov elements of the mapping class group of an oriented surface from the point of view of fibered face theory. This theory runs dual to the classical way of thought where, rather than fixing a surface S and studying Mod(S), we study the set of all pseudoAnosov mapping classes over all oriented surfaces. Let S be a connected, compact, oriented surface. The mapping class group of S, denoted Mod(S), is the group of orientation preserving homeomorphisms of S...
Show moreThis dissertation explores pseudoAnosov elements of the mapping class group of an oriented surface from the point of view of fibered face theory. This theory runs dual to the classical way of thought where, rather than fixing a surface S and studying Mod(S), we study the set of all pseudoAnosov mapping classes over all oriented surfaces. Let S be a connected, compact, oriented surface. The mapping class group of S, denoted Mod(S), is the group of orientation preserving homeomorphisms of S which act by the identity on θS considered up to isotopy. If no power of a mapping class leaves an essential curve invariant, the mapping class is pseudoAnosov. In this case, the mapping class preserves an expanding and contracting foliation with expansion factor λ. The set of all pseudoAnosov mapping classes admits a natural partition into flow equivalence classes. Such a class can be described as the surface cross sections transverse to a pseudoAnosov flow in a hyperbolic fibered three manifold. Using the operation of Murasugi sum, we systematically study the flow equivalence classes that can be expressed as iterated Hopf plumbings on a disk in the 3sphere. Such a surface is always a fiber surface for its boundary link. The data on how to attach the Hopf bands is conveniently packaged in a graph and, since the Coxeter element of this graph is, up to sign, the monodromy of the fiber surface, such links are called Coxeter links. The investigation splits into three main developments. The first result deals with the overall structure of the flow equivalence classes corresponding to Coxeter links as subspaces of the real vector space H¹(M;[the set of real numbers]), where M is the link exterior in the 3sphere. The second result sheds light on a natural dynamically minimal representative in each class. We then give an algorithm that takes a class as input and outputs a multivariable polynomial which can be used to compute the expansion factor of any element contained in the class. By interpreting the mapping tori of the pseudoAnosov mapping classes as link exteriors in the 3sphere, we are able to identify the meridians of the link components with a basis for H¹(M;[integers]). With a few careful knot theoretic observations, we show that any surface with positive linking number to the original link is a fiber surface. With slightly stronger assumptions on the link, we show that the entire Thurston norm is determined by the norms of spanning surfaces for the individual components. It is easy to construct pseudoAnosov mapping classes with small expansion factor on surfaces with high Euler characteristic. One way this can be achieved is by composing a periodic mapping class with a pseudoAnosov map that is supported on a small subsurface. Since the flow equivalence class of a pseudoAnosov homeomorphism contains maps supported on surfaces of arbitrarily high Euler characteristic, we consider the function λ❘x(S)❘. Using properties of this function and the above results, we find a natural minimizing element with respect to this function. The third result amounts to computing the Teichmüller polynomial for the fibered face in question. This can be a difficult process in general. Perhaps the most notable issues are explicitly computing the fixed cohomology and a traintrack for a surface automorphism. After finding ways around these problems and others, we give the full algorithm to compute the Teichmüller polynomial.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Billet_fsu_0071E_13563
 Format
 Thesis
 Title
 Hypergeometric Solutions of Linear Differential Equations with Rational Function Coefficients.
 Creator

Kunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State...
Show moreKunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State University
Show less  Abstract/Description

Let L be a second order linear differential equation with rational function coefficients. We want to find a solution (if that exists) of L in terms of 2F1hypergeometric function. This thesis presents two algorithms to find such solution in the following cases: 1. L has five regular singularities where at least one of them is logarithmic. 2. L has hypergeometric solution of degree three, i.e, L is solvable in terms of 2F1(a,b;c  f) where f is a rational function of degree three.
 Date Issued
 2014
 Identifier
 FSU_migr_etd9021
 Format
 Thesis
 Title
 On Elliptic Fibrations and FTheory Compactifications of String Vacua.
 Creator

Fullwood, James, Aluﬃ, Paolo, Reina, Laura, Van Hoeij, Mark, Aldrovandi, Ettore, Hironaka, Eriko, Department of Mathematics, Florida State University
 Abstract/Description

We investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible...
Show moreWe investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible dimensions of a base at once. As such, we construct generating series $f(t)$ corresponding to each invariant such that the coefficient of $t^k$ encodes the invariant of the elliptic fibration over a base of dimension $k$, solely in terms of invariants of the base. From the Ftheory perspective, we highlight aspects of elliptic fibrations other than Weierstrass models, and construct a new orientifold limit of Ftheory associated with $D_5$ fibrations, i.e., elliptic fibrations whose elliptic fiber is realized via a complete intersection of two quadrics in $\mathbb{P}^3$. We verify tadpole relations as predicted by the (conjectural) equivalence between Ftheory and typeIIB, as well as `universal tadpole relations', which are mathematical generalizations of the tadpole relations predicted by the physics of Ftheory. We also simplify formulas for invariants of CalabiYau fourfolds, and suggest that all Hodge numbers of CalabiYau fourfolds depend linearly on $c_1(B)^3$, where $B$ is the base of the fibration.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4848
 Format
 Thesis
 Title
 Periodic Pieces of PseudoAnosov Flows in Graph Manifolds.
 Creator

Waller, Russell L., Fenley, Sérgio Roberto, Reina, Laura, Hironaka, Eriko, Heil, Wolfgang H., Mio, Washington, Florida State University, College of Arts and Sciences, Department...
Show moreWaller, Russell L., Fenley, Sérgio Roberto, Reina, Laura, Hironaka, Eriko, Heil, Wolfgang H., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Our work builds on that of Barbot and Fenley to generalize Bonatti and Langevin's famous construction of a graph manifold with pseudoAnosov flow in which all Seiftert fibered pieces of the torus decomposition are periodic. We provide infinitely many new examples of such graph manifolds, as well as a complete classification  up to Seifert invariant  in the case that each Seiftert fibered piece is orientable and the flow is Anosov. We further demonstrate that the singularities of the flow...
Show moreOur work builds on that of Barbot and Fenley to generalize Bonatti and Langevin's famous construction of a graph manifold with pseudoAnosov flow in which all Seiftert fibered pieces of the torus decomposition are periodic. We provide infinitely many new examples of such graph manifolds, as well as a complete classification  up to Seifert invariant  in the case that each Seiftert fibered piece is orientable and the flow is Anosov. We further demonstrate that the singularities of the flow are not rigid but can rather be "rearranged", or even removed, without affecting the ambient manifold. To build our graph manifolds and model the pseudoAnosov flows that they support, we define and construct combinatorial objects known as flow graphs. We study these flow graphs and the surfaces, called fat graphs (or ribbon graphs), that retract onto them. In particular, we study flow graphs with the additional conditions needed to generate pseudoAnosov flows from the combinatorial data that the flow graphs provide, and classify the surfaces that admit flow graphs with these additional requirements.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9479
 Format
 Thesis
 Title
 Principal Elements of MixedSign Coxeter Systems.
 Creator

Armstrong, Johnathon Kyle, Hironaka, Eriko, Petersen, Kathleen, Chicken, Eric, Aldrovandi, Ettore, Bellenot, Steven, Van Hoeij, Mark, Department of Mathematics, Florida State...
Show moreArmstrong, Johnathon Kyle, Hironaka, Eriko, Petersen, Kathleen, Chicken, Eric, Aldrovandi, Ettore, Bellenot, Steven, Van Hoeij, Mark, Department of Mathematics, Florida State University
Show less  Abstract/Description

In this thesis we generalize results from classical Coxeter systems to mixedsign Coxeter systems which are denoted by a triple (W,S,B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a bilinear form Bon R n. A generator s i in the set S is defined to negate the ith basis vector of R n and fix the set of vectors v which are orthogonal relative to B. Classical Coxeter theory works in this fashion, here we generalize this notion to encompass both...
Show moreIn this thesis we generalize results from classical Coxeter systems to mixedsign Coxeter systems which are denoted by a triple (W,S,B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a bilinear form Bon R n. A generator s i in the set S is defined to negate the ith basis vector of R n and fix the set of vectors v which are orthogonal relative to B. Classical Coxeter theory works in this fashion, here we generalize this notion to encompass both Coxeter systems in addition to mixedsign Coxeter systems. As in classical Coxeter theory, we show that the bilinear form may be used to compute an element of the reflection group called a principal element. In classical Coxeter groups, the principal elements have been shown to have special properties. The socalled deletion condition is a property of classical Coxeter systems which allows Coxeter groups to have a presentation which only depends on pairwise relationships between generators. Here, we show that mixedsign Coxeter systems do not generally have the deletion condition. We give a correspondence between a graph $\Gamma$ and the reflection system (W,S,B). We refer to the reflection group associated to &Gamma by W (&Gamma). We show an isomorphism of mixedsign Coxeter groups; explicitly if &Gamma is a bipartite mixedsign Coxeter graph and &Gamma is the mixedsign Coxeter graph with all the nodes of &Gamma negated then (W,S,B(&Gamma)) and (W,S,B(&Gamma)) are conjugate reflection systems. Furthermore, we indicate the the bipartite condition is necessary. We show a class of examples; odd cycles with all negative nodes where negating all the nodes gives a reflection system which is not conjugate. Additionally, we show that the spectral radius of mixedsign Coxeter elements are not bounded below by the bipartite eigenvalue of the mixedsign Coxeter system, this is another distinguishing feature of mixedsign Coxeter systems from their classical counterparts and provides an interesting avenue of research to pursue in the future.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4697
 Format
 Thesis
 Title
 Sequences of PseudoAnosov Mapping Classes with Asymptotically Small Dilatation.
 Creator

Valdivia, Aaron David, Hironaka, Eriko, Reina, Laura, Heil, Wolfgang, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

We construct sequences of pseudoAnosov examples which we use to bound the minimal dilatation on arbitrary surfaces. We show that these bounds give the asymptotic behavior of the minimal dilatations for certain sequences. Further we show that the mapping classes for a given sequence from our construction can be realized as fibrations of a single 3manifold.
 Date Issued
 2011
 Identifier
 FSU_migr_etd5242
 Format
 Thesis
 Title
 Topology of ngonal Curve Complements.
 Creator

Aktas, Mehmet Emin, Hironaka, Eriko, Mio, Washington, Kumar, Piyush, Heil, Wolfgang, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of...
Show moreAktas, Mehmet Emin, Hironaka, Eriko, Mio, Washington, Kumar, Piyush, Heil, Wolfgang, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis has two parts. The first part concerns topological invariants of the ngonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group. Our second results is an algorithm to compute the braid monodromy and Libgober polynomial invariant of ngonal curves. We show that the algorithm improves on existing algorithms. We compare the...
Show moreThis thesis has two parts. The first part concerns topological invariants of the ngonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group. Our second results is an algorithm to compute the braid monodromy and Libgober polynomial invariant of ngonal curves. We show that the algorithm improves on existing algorithms. We compare the information one gets from Alexander and Krammer polynomials. The second and main part of our thesis focuses on properties of dessins d'enfants associated to trigonal curves. Degtyarev first studied dessins d'enfants in this context giving a new method for computing braid monodromies and fundamental groups. Our first result is a classification of all possible combinatorial data that can occur for trigonal curves of low degree, as well as bounds on the number of possibilities for all degree. We also study deformations of trigonal curves and corresponding deformations of their dessins. Of special interest to Degtyarev was the case when the dessins are maximal. Our second result gives a sufficient condition for a trigonal curve to be deformable to one that is maximal.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Aktas_fsu_0071E_13779
 Format
 Thesis