Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Torsion of Noncircular Members01:16

Torsion of Noncircular Members

Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...
Toroids01:27

Toroids

A toroid is a closely wound donut-shaped coil constructed using a single conducting wire. In general, it is assumed that a toriod consists of multiple circular loops perpendicular to its axis.
When connected to a supply, the magnetic field generated in the toroid has field lines circular and concentric to its axis. Conventionally, the direction of this magnetic field is expressed using the right-hand rule. If the fingers of the right hand curl in the current direction, the thumb points in the...
Torsion in Vector Calculus01:20

Torsion in Vector Calculus

A toy train ascending a winding track that curves and tilts offers an intuitive view of torsion, a key geometric concept in the study of space curves. While curvature measures how sharply a path bends, torsion captures how the path twists out of the plane of bending. This twisting behavior is crucial in understanding three-dimensional motion and is precisely described using the Frenet–Serret framework.At each point along a space curve, the Frenet–Serret frame consists of three orthogonal unit...
Quadric Surfaces01:28

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...
Mohr's Circle for Plane Strain01:18

Mohr's Circle for Plane Strain

Mohr's circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain ∈ and the ordinate is shear strain γ. Similarly to Mohr’s circle for plane stress, two points X and Y are plotted. Their coordinates are (∈x, -γXY) and (∈Y, γXY), respectively.
Mohr's circle visually represents the strain states under various conditions, which is essential for understanding material behavior. The center of Mohr's...
Solid–Solid Solutions01:24

Solid–Solid Solutions

The temperature-composition phase diagram of two solids, A and B, which are immiscible in the solid phase but form miscible liquids, shows that when the temperature is low, these two exist as separate, pure solids (A and B). As the temperature increases, they transition into a single-phase liquid solution where A and B coexist. Moving from point a1 to a2 in the phase diagram, the composition changes such that solid B begins to separate from the solution, enriching the remaining liquid with A.

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

In This Issue.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Long-term cultural continuity across the Neanderthal-modern human sequence at Üçağızlı II Cave, northern Levant.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Dolphins use names to remember whom to avoid.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Retraction for Shaked and Frenkel, Curiouser and curiouser: Meningeal lymphoid structures in the aging brain.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Small but mighty: The outsized role of small water bodies in the global carbon cycle.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Functional traits produce conditional outcomes in different community contexts.

Proceedings of the National Academy of Sciences of the United States of America·2026
See all related articles

Related Experiment Video

Updated: Jun 28, 2026

Rotating the Intraocular Lens to Prevent Posterior Capsular Opacification in Cataract Surgeries
04:59

Rotating the Intraocular Lens to Prevent Posterior Capsular Opacification in Cataract Surgeries

Published on: July 7, 2023

Knotted solid tori in contact manifolds.

John B Etnyre1, Youlin Li2, Bülent Tosun3

  • 1School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332.

Proceedings of the National Academy of Sciences of the United States of America
|June 26, 2026
PubMed
Summary
This summary is machine-generated.

This study reveals that "nonthickenable" tori are common in knot theory, not rare as previously thought. These findings are crucial for understanding contact manifolds and knot invariants.

Keywords:
Legendrian knotscontact geometryknotted tori

More Related Videos

Novel Triple-Loop Technique for Suturing TFCC Injuries without Transosseous Tunnel
08:27

Novel Triple-Loop Technique for Suturing TFCC Injuries without Transosseous Tunnel

Published on: May 23, 2025

Related Experiment Videos

Last Updated: Jun 28, 2026

Rotating the Intraocular Lens to Prevent Posterior Capsular Opacification in Cataract Surgeries
04:59

Rotating the Intraocular Lens to Prevent Posterior Capsular Opacification in Cataract Surgeries

Published on: July 7, 2023

Novel Triple-Loop Technique for Suturing TFCC Injuries without Transosseous Tunnel
08:27

Novel Triple-Loop Technique for Suturing TFCC Injuries without Transosseous Tunnel

Published on: May 23, 2025

Area of Science:

  • Topology
  • Differential Geometry
  • Contact Geometry

Background:

  • Solid tori are fundamental in contact manifold studies.
  • The contact width of a knot type is a key invariant.
  • Nonthickenable tori were previously thought to be rare, observed only for specific knot types.

Purpose of the Study:

  • To investigate the prevalence of nonthickenable tori in contact manifolds.
  • To establish criteria for computing the contact width of knot types.
  • To explore the significance of these tori in Legendrian and transverse knot theory.

Main Methods:

  • Analysis of solid tori within contact manifolds.
  • Development of criteria for explicit computation of contact width.
  • Demonstration of the existence of nonthickenable tori in various knot types and manifolds.

Main Results:

  • Criteria for computing contact width are established.
  • A significant number of nonthickenable tori are proven to exist in many knot types.
  • Nonthickenable tori are shown to be common, existing in diverse manifolds, including for hyperbolic knots.

Conclusions:

  • Nonthickenable tori are more prevalent than previously assumed.
  • The findings impact the study of tight contact structures and knot invariants.
  • New conjectures regarding knots in contact manifolds are proposed.