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Articles 1 - 9 of 9
Full-Text Articles in Geometry and Topology
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 …
Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger
Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger
Dissertations, Theses, and Capstone Projects
This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, JTn, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show JTn satisfies many properties of the original colored Jones polynomial. Most significantly, JTn exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions …
Linear Progress With Exponential Decay In Weakly Hyperbolic Groups, Matthew H. Sunderland
Linear Progress With Exponential Decay In Weakly Hyperbolic Groups, Matthew H. Sunderland
Dissertations, Theses, and Capstone Projects
A random walk wn on a separable, geodesic hyperbolic metric space X converges to the boundary ∂X with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on X is acylindrical. We extend exponential decay to the nonacylindrical case. We give an application to random Heegaard splittings.
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.
Torsion In One-Term Distributive Homology, Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra
Torsion In One-Term Distributive Homology, Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra
Mathematics, Statistics and Data Science Faculty Works
The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this paper that the one-term homology of a finite spindle can have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, …
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.