Related papers: Hidden symmetries via hidden extensions
We establish a pair of criteria for proving that most knot complements obtained as Dehn fillings of a given two-component hyperbolic link complement lack hidden symmetries. To do this, we use certain rational functions on varieties…
In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(\epsilon, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist…
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based…
Motivated by a question of Neumann and Reid, we study whether Dehn fillings on all but one cusp of a hyperbolic link complement can produce infinite families of knot complements with hidden symmetries which geometrically converge to the…
This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one…
We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the…
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic…
The extension of the knot group $\pi_1(S^3\setminus K)$ to the category of tangles is introduced via a new category-theoretic construction. Through this presentation, a new avenue of proof for results about knot groups is opened.
Hidden symmetries, described by higher order in momenta integrals of motion that generate nonlinear algebras, are explored at the level of classical and quantum mechanics in a variety of physical systems related to conformal and…
We classify all finite group actions on knots in the 3-sphere. By geometrization, all such actions are conjugate to actions by isometries, and so we may use orthogonal representation theory to describe three cyclic and seven dihedral…
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and…
Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to…
This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture…
We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer…
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other…
We consider knots and links in handlebodies that have hyperbolic complements and operations akin to composition. Cutting the complements of two such open along separating twice-punctured disks such that each of the four resulting…
We give a formalism for constructing hidden sector bundles as extensions of sums of line bundles in heterotic $M$-theory. Although this construction is generic, we present it within the context of the specific Schoen threefold that leads to…
We present new techniques to show hyperbolicity of links based on geometric/combinatorial topology. Our techniques are applicable to links that have at least one unknotted component. In particular, they are applicable to Brunnian links. We…
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle…
We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the…