Narrow Results

In this paper, we define a new algebro-geometric invariant of three-manifolds resulting from Dehn surgery along a hyperbolic knot complement in S³. We establish a Casson-type invariant for these three-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some three-manifolds resulting from Dehn surgery along the figure-eight knot.

Let n > m > 2 be two fixed coprime integers. We prove that two Conway reducible, hyperbolic knots sharing the 2-fold, m-fold and n-fold cyclic branched covers are equivalent. Using previous results by Zimmermann we prove that this implies that a hyperbolic knot is determined by any three of its cyclic branched covers.

Recently Kearton showed that any Seifert matrix of a knot is S-equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S-equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.

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.

Based on symbolic dynamics of Lorenz maps, we prove that, provided one conjecture due to Morton is true, then Lorenz knots associated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type $(X,Y)\ast S$, where the sequences $X$ and $Y$ are Farey neighbors verifying some conditions, are hyperbolic.

For a hyperbolic link $K$ in the thickened torus, we show there is a decomposition of the complement of a link $L$, obtained from augmenting $K$, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [D. Futer and F. Guéritaud, From angled triangulations to hyperbolic structures, in Interactions between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporary Mathematics, Vol. 541 (American Mathematical Society, Providence, RI, 2011), pp. 159–182.], this shows that $L$ is hyperbolic.

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL₂(ℂ) character variety of the hyperbolic knot in S³. Furthermore, we prove that the corresponding ℂ*-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over ℂ* × ℂ*. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.