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Jessica S. Purcell

Publications and source records attributed to Jessica S. Purcell.

71 records · Page 4Linked to original sources
arXiv2012

Guts of surfaces and the colored Jones polynomial

This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A- or B-adequacy), we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses (A- or B-adequacy), we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We employ normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between quantum and geometric knot invariants.

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arXiv2012

Jones polynomials, volume, and essential knot surfaces: a survey

This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [arXiv:1002.0256] and [arXiv:1108.3370], while this survey focuses on the main ideas and examples.

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arXiv2012

Volumes of chain links

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n at most 10. In his thesis, Venzke mentions that these cannot be smallest volume for n at least 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n at least 60 we use a formula from work of Futer, Kalfagianni, and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m+1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n is at least 60 or |m| is at least 8, and we give computational data indicating this remains true for smaller n and |m|.

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arXiv2011

Volume bounds for generalized twisted torus links

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 the number of strands being twisted. We exhibit a family of twisted torus knots for which this upper bound is sharp, and another family with volumes approaching infinity. Consequently, we show there exist twisted torus knots with arbitrarily large braid index and yet bounded volume.

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arXiv2010

Slopes and colored Jones polynomials of adequate knots

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 of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.

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arXiv2010

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 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture.

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arXiv2009

Hyperbolic geometry of multiply twisted knots

We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based only on a diagram. The results are obtained by finding geometric information on generalized augmentations of these links.

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arXiv2009

On multiply twisted knots that are Seifert fibered or toroidal

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized augmented links.

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arXiv2009

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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arXiv2009

Geometric limits of knot complements

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3--manifold with two convex cocompact ends cannot be a geometric limit of knot complements in $\BS^3$.

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arXiv2009

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and $λ_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or $λ_1$. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on $λ_1$. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.

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arXiv2008

Cusp shapes under cone deformation

A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighborhood of the singular locus alone. We then apply this result to provide information on the cusp shape of many hyperbolic knots, given only a diagram of the knot. Specifically, we show there is a universal constant C such that if a knot admits a prime, twist reduced diagram with at least C crossings per twist region, then the length of the second shortest curve on the cusp torus is bounded. The bound is linear in the number of twist regions of the diagram.

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arXiv2008

Slope lengths and generalized augmented links

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from below. Finally, we present an example to show that, in contrast to the case of the regular augmented link, longitude lengths of these knots cannot be determined by a function of the number of twist regions alone.

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arXiv2008

Dehn filling, volume, and the Jones polynomial

Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

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arXiv2007

Volumes of highly twisted knots and links

We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113.

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arXiv2005

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. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.

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