Related papers: Knot complements, hidden symmetries and reflection…
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these…
Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…
It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a…
A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called…
It is well-known that complex hyperbolic triangle groups $\Delta(3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mbox{PU(2,1)}$ has 1-dimensional moduli space. Deforming the representations from the classical…
We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.
We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…
We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. We show that, under mild hypothesis, their cusp area admits two sided bounds in terms of the twist number of the…
We prove that for three-manifolds satisfying a certain algebraic condition on their fundamental group, null-homotopic knots are determined by their complements. This answers a Kirby Problem posed by Boileau for this special case of…
We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger…
Given any knot k, there exists a hyperbolic knot tilde k with arbitrarily large volume such that the knot group pi k is a quotient of pi tilde k by a map that sends meridian to meridian and longitude to longitude. The knot tilde k can be…
It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.
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…
The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere…
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot…
In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under…
Given any knot K in the 3-sphere, we prove that there are only finitely many hyperbolic fibered knots which are ribbon concordant to K. It follows that every fibered knot in the 3-sphere has only finitely many hyperbolic predecessors under…
In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher,…
It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed three manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum…
We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight knot, the homology 3-sphere defined by the Kirby diagram which is a simple link of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead double…