The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its… (More)
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by… (More)
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these " simple " hyperbolic knots have high crossing number. We also compute their Jones polynomials.
We show that the Mahler measure of the Jones polynomial and of the colored Jones polynomials converges under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial, like the Alexander polynomial, behaves like hyperbolic volume under… (More)
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás–Riordan–Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning… (More)
The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.
Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C×C which contains the squares of eigenvalues of PSL(2, C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related… (More)
Given any knot k, there exists a hyperbolic knot˜k with arbitrarily large volume such that the knot group πk is a quotient of π ˜ k by a map that sends meridian to meridian and longitude to longitude. The knot˜k can be chosen to be ribbon concordant to k and also to have the same Alexander invariant as k. 1. Introduction. The classical problem of topology… (More)
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their hyperbolic volumes that depend only on… (More)