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

Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

499
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
499
SFG Algebra01:16

SFG Algebra

467
In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
467
Singularity Functions for Shear01:26

Singularity Functions for Shear

545
In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the shear...
545
Rolle’s Theorem01:09

Rolle’s Theorem

222
Rolle’s Theorem states that if a real-valued function is continuous on a closed interval, differentiable on the open interval, and takes equal values at both endpoints, then there is at least one point within the open interval where the derivative of the function is zero.Rolle’s Theorem describes an important property of differentiable functions, this theorem applies to a real-valued function defined on a closed interval, provided three specific conditions are met. First, the...
222
Vector Algebra: Graphical Method01:10

Vector Algebra: Graphical Method

13.7K
Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
We use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at arbitrary positions. Translate either one of...
13.7K
Castigliano's Theorem01:18

Castigliano's Theorem

1.4K
Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.
1.4K

You might also read

Related Articles

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

Sort by
Same journal

Correction for Chang et al., Plant pathogenic nematode exosomes remodel vector tracheae to enhance pathogen transmission.

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

Correction for Cao et al., Cenozoic geoclimatic changes drove the evolutionary dynamics of floristic endemism on the Qinghai-Tibet Plateau.

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

Hydration and hydrolysis define antibiotic resistance conferred by macrolide esterases.

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

Alpha rhythms in the left auditory cortex set the speed limit for speech comprehension.

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

A population of primary afferent sensory neurons mediates pain relief through nocifensive coping behavior in mice.

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

Evidence of intertwined pair density and charge density wave orders in UTe<sub>2</sub>.

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

Related Experiment Video

Updated: Apr 27, 2026

Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser
09:00

Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser

Published on: June 28, 2018

9.3K

Positive basis for surface skein algebras.

Dylan Paul Thurston1

  • 1Department of Mathematics, Indiana University, Bloomington, IN 47405 dpthurst@indiana.edu.

Proceedings of the National Academy of Sciences of the United States of America
|July 2, 2014
PubMed
Summary

We discovered a positive basis, called the bracelets basis, for the twisted SL2 skein algebra of a surface. This basis simplifies algebraic structures by ensuring multiplication constants are positive integers.

Keywords:
Jones polynomialcanonical basischaracter varietycluster algebras

More Related Videos

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces
06:14

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces

Published on: September 11, 2018

6.1K
Functional Surface-immobilization of Genes Using Multistep Strand Displacement Lithography
11:05

Functional Surface-immobilization of Genes Using Multistep Strand Displacement Lithography

Published on: October 25, 2018

7.0K

Related Experiment Videos

Last Updated: Apr 27, 2026

Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser
09:00

Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser

Published on: June 28, 2018

9.3K
Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces
06:14

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces

Published on: September 11, 2018

6.1K
Functional Surface-immobilization of Genes Using Multistep Strand Displacement Lithography
11:05

Functional Surface-immobilization of Genes Using Multistep Strand Displacement Lithography

Published on: October 25, 2018

7.0K

Area of Science:

  • Algebraic Topology
  • Knot Theory
  • Surface Algebra

Background:

  • The study of skein algebras is crucial in understanding topological invariants.
  • The twisted SL2 skein algebra is a significant structure in low-dimensional topology.

Purpose of the Study:

  • To identify and characterize a natural basis for the twisted SL2 skein algebra of a surface.
  • To demonstrate the positivity property of this basis.

Main Methods:

  • Introduction of the 'bracelets basis' for the twisted SL2 skein algebra.
  • Analysis of the multiplication structure within this basis.

Main Results:

  • The bracelets basis is shown to be a natural basis for the twisted SL2 skein algebra.
  • The structure constants for multiplication in the bracelets basis are positive integers, confirming its positivity.

Conclusions:

  • The discovery of a positive basis simplifies the study of twisted SL2 skein algebras.
  • This finding has implications for algebraic topology and related fields.