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Articles 1 - 19 of 19
Full-Text Articles in Geometry and Topology
Numerical Investigation To Produce A Fundamental Polygon, Elizabeth Sipes
Numerical Investigation To Produce A Fundamental Polygon, Elizabeth Sipes
Honors College Theses
There exist multiple types of geometry, differing in the postulates they are based on, and therefore the theorems and proofs that make up said geometry. Hyperbolic geometry differs from others by allowing there to exist multiple lines through a single point not on a given line, that are parallel to the given line. Every geometry has the idea of distance and isometries, distance preserving maps. By considering special collections of isometries called discrete groups, we can construct interesting surfaces, such as the torus and genus-g surface. The connection between the surface and the discrete group can be understood through …
Farey Recursion And Hyperbolic Dehn Filling, Jose Ebenezer Martinez
Farey Recursion And Hyperbolic Dehn Filling, Jose Ebenezer Martinez
Graduate Student Theses, Dissertations, & Professional Papers
In this work, we present a solution to William Thurston's edge gluing equations for Dehn fillings of hyperbolic 3-manifolds. This is done for triangulations that involve the layered solid torus. Our approach uses Farey recursive functions, and we present a Farey recursive function that provides a solution to the gluing equations for any hyperbolic Dehn filling admitting a triangulation by the layered solid torus. We provide examples that demonstrate our solution for multiple 3-manifolds, and study the roots of the corresponding Farey recursive polynomials. As an additional application of our solution, we provide a formula for the complex length of …
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu
Rose-Hulman Undergraduate Mathematics Journal
In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.
Constructing Hyperbolic Polygons In The Poincaré Disk, Akram Zakaria Samweil
Constructing Hyperbolic Polygons In The Poincaré Disk, Akram Zakaria Samweil
Electronic Theses, Projects, and Dissertations
The Poincaré Disk plays a significant role in non-Euclidean geometry. Inverting points, segments, or polygons through a circle provides us with a deep vision of the link between Euclidean and non-Euclidean geometry; especially when we try to prove lemmas, constructions, or conjectures. All points inside a circle, c, represent a Poincaré Disk denoted Dc, and all "lines" in a Poincaré Disk are d-lines, which are circles orthogonal to the circle's boundary. The question that motivated my research is: how can we use the properties of the disk, its boundary, and its d-lines to construct hyperbolic polygons? We will …
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
Electronic Theses, Projects, and Dissertations
This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i2 = 3, j2 = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL2(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …
Differentiability Of The Liouville Map Via Geodesic Currents, Xinlong Dong
Differentiability Of The Liouville Map Via Geodesic Currents, Xinlong Dong
Dissertations, Theses, and Capstone Projects
For a conformally hyperbolic Riemann surface, the Teichmüller space is the space of quasiconformal maps factored by an equivalence relation, and it is a complex Banach manifold. The space of geodesic currents endowed with the uniform weak* topology is a subset of a Fréchet space of Hölder distributions. We introduce an appropriate topology on the space of Hölder distributions and this new topology coincides with the uniform weak* topology on the space of geodesic currents. The Liouville map of the Teichmüller space becomes differentiable in the Fréchet sense. In particular, the derivative of Liouville currents exists and belongs to the …
Visualizing Geometric Structures On Topological Surfaces, Andrea Clark
Visualizing Geometric Structures On Topological Surfaces, Andrea Clark
All NMU Master's Theses
We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the …
Growth Of Conjugacy Classes Of Reciprocal Words In Triangle Groups, Blanca T. Marmolejo
Growth Of Conjugacy Classes Of Reciprocal Words In Triangle Groups, Blanca T. Marmolejo
Dissertations, Theses, and Capstone Projects
In this thesis we obtain the growth rates for conjugacy classes of reciprocal words for triangle groups of the form G = Z2 ∗ H where H is finitely generated and does not contain an order 2 element. We explore cases where H is infinite cyclic and finite cyclic. The quotient O = H/G is an orbifold and contains a cone point of order 2, due to the first factor Z2 in the free product G. The reciprocal words in G correspond to geodesics on O which pass through the order 2 cone point on O. We use methods from …
Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan
Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan
Mathematics, Statistics and Data Science Faculty Works
We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan
Mathematics, Statistics and Data Science Faculty Works
In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan
Mathematics, Statistics and Data Science Faculty Works
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.
Upper Bounds In The Ohtsuki-Riley-Sakuma Partial Order On 2-Bridge Knots, Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan
Upper Bounds In The Ohtsuki-Riley-Sakuma Partial Order On 2-Bridge Knots, Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan
Mathematics, Statistics and Data Science Faculty Works
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K we characterize all other 2-bridge knots J such that {K, J} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.
Commensurability Classes Of Twist Knots, Jim Hoste, Patrick D. Shanahan
Commensurability Classes Of Twist Knots, Jim Hoste, Patrick D. Shanahan
Mathematics, Statistics and Data Science Faculty Works
In this paper we prove that if MK is the complement of a non-fibered twist knot K in S3, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Tilings Of Low-Genus Surfaces By Quadrilaterals, John Gregoire, Isabel Averil
Mathematical Sciences Technical Reports (MSTR)
In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods
Mathematical Sciences Technical Reports (MSTR)
The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL2(13). These tilings can be distinguished by the lengths of their systoles.
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter
Mathematical Sciences Technical Reports (MSTR)
The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Constructing Kaleidscopic Tiling Polygons In The Hyperbolic Plane, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see [6].
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Symmetry And Tiling Groups For Genus 4 And 5, C. Ryan Vinroot
Mathematical Sciences Technical Reports (MSTR)
All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Quadrilaterals Subdivided By Triangles In The Hyperbolic Plane, Dawn M. Haney, Lori T. Mckeough
Mathematical Sciences Technical Reports (MSTR)
In this paper, we consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. These tilings are called divisible tilings or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertexof the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface on which the divisible tiling exists. The tiling groups of these surfaces are also determined.