Let T WD .RXZ/=Z be the once-punctured torus endowed with its differential structure and an orientation. The group G of isotopy classes of orientation-preserving diffeomorphisms 'W T ! T (or theâ€¦ (More)

This work initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariantsâ€¦ (More)

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence ofâ€¦ (More)

This paper continues our study, initiated in [13], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we showâ€¦ (More)

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. Weâ€¦ (More)

This paper gives the first explicit, twoâ€“sided estimates on the cusp area of onceâ€“ punctured torus bundles, 4â€“punctured sphere bundles, and 2â€“bridge link complements. The input for these estimates isâ€¦ (More)

This survey paper contains an elementary exposition of Casson and Rivinâ€™s technique for finding the hyperbolic metric on a 3â€“manifold M with toroidal boundary. We also survey a number of applicationsâ€¦ (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 alsoâ€¦ (More)

Let Ip,v be Bourdonâ€™s building, the unique simply-connected 2â€“complex such that all 2â€“cells are regular right-angled hyperbolic pâ€“gons and the link at each vertex is the complete bipartite graphâ€¦ (More)

In recent years, several families of hyperbolic knots have been shown to have both volume and Î»1 (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while otherâ€¦ (More)