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- Abhijit Champanerkar, Ilya Kofman, +4 authors Neal W. Stoltzfus
- 2007

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)

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)

- Abhijit Champanerkar, Ilya Kofman, Jessica S. Purcell
- J. London Math. Society
- 2016

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating… (More)

Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing… (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)

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)

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)

- Daniel S. Silver, Wilbur Whitten, Abhijit Champanerkar, Tim Cochran, Danny Ruberman
- 2005

Given any knot k , there exists a hyperbolic knot k̃ with arbitrarily large volume such that the knot group πk is a quotient of πk̃ by a map that sends meridian to meridian and longitude to longitude. The knot k̃ can be chosen to be ribbon concordant to k and also to have the same Alexander invariant as k . AMS Classification 57M25; 20F34

Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.

It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that… (More)