Narrow Results

In this paper we compute the trace field for the family of hyperbolic twist knots. We describe this field as a simple extension ℚ(z₀) where z₀ is a specified root of a particular irreducible polynemial Φ_n(z)∈ℤ[z]. As a consequence, we find that the degree of the trace field is precisely two less than the-minimal crossing number of a twist knot.

For a hyperbolic knot K in S³, a toroidal surgery on K is integral or half-integral. It is well known that no 2-bridge knot admits non-integral toroidal surgery. In this paper, we show that the (-2, 3, 7)-pretzel knot and its mirror image are the only 3-bridge knots that admit non-integral toroidal surgery.

It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding non-hyperbolike 3-manifolds; namely, 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.

In this paper, we give a formula of the complex volumes of hyperbolic knot complements, which is related to the volume conjecture for hyperbolic knots.

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.

In this paper, we give a formula of the cusp shape of hyperbolic knots by using potential functions which appears in the study of the volume conjecture.

For each positive integer $n$ we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to $n$, braid index equal to $n+3$ and Turaev genus equal to $n$.

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such hyperbolic knots and posed a conjecture that his examples encompass all hyperbolic knots in lens spaces with non-prime surgeries. Using the idea of seiferters we construct a counterexample to this conjecture. In the process of construction, we also derive an obstruction for a small Seifert fibered space to be obtainable by a surgery with a seiferter.

In this paper we construct an infinite family of hyperbolic knots, each having a Dehn surgery which produces a manifold containing an incompressible torus, which hits the core of the surgered solid torus in four points, but containing no incompressible torus hitting it in less than four points.