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Guts of Surfaces and the Colored Jones Polynomial
TLDR
We introduce new invariants for links in 3–manifolds, as well as invariants of hyperbolic knots. Expand
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Geometrically and diagrammatically maximal knots
TLDR
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 for the knot determinant per crossing. Expand
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An introduction to fully augmented links
In this article we summarize information on the class of fully augmented links. These links are geometrically explicit, and therefore provide a large class of examples of hyperbolic links for whichExpand
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Links with no exceptional surgeries
We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike.Expand
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Geometry of alternating links on surfaces.
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising theExpand
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Geometry of biperiodic alternating links
TLDR
In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling, for an analogous notion of a polyhedral decomposition. Expand
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The length of unknotting tunnels
We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.
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Cusp Areas of Farey Manifolds and Applications to Knot Theory
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere bundles, and two-bridge link complements. The input for these estimatesExpand
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Volume bounds for weaving knots
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 firstExpand
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Quasifuchsian state surfaces
This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolicExpand
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