scholarly journals The Khovanov homology of knots

2021 ◽  
Author(s):  
◽  
Giovanna Le Gros
Keyword(s):  
Knot Theory ◽  
Homological Algebra ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Kauffman Bracket ◽  
Knot Invariant ◽  
Definition Of

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

scholarly journals The Khovanov homology of knots

2021 ◽  
Author(s):  
◽  
Giovanna Le Gros
Keyword(s):  
Knot Theory ◽  
Homological Algebra ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Kauffman Bracket ◽  
Knot Invariant ◽  
Definition Of

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

scholarly journals AN ORIENTED MODEL FOR KHOVANOV HOMOLOGY

2010 ◽  
Vol 19 (02) ◽  
pp. 291-312 ◽  
Author(s):  
CHRISTIAN BLANCHET
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
State Model ◽  
Boundary Operator ◽  
Gaussian Integers ◽  
Kauffman Bracket ◽  
Trivalent Graphs ◽  
Definition Of ◽  
Original Construction

We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers.

scholarly journals Enumeration on graph mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Research Program ◽  
Knot Theory ◽  
Quantum Physics ◽  
Recursion Formula ◽  
Jones Polynomial ◽  
Exact Enumeration ◽  
State Matrix ◽  
Embedded Graph ◽  
Definition Of

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

scholarly journals KHOVANOV HOMOLOGY AND THE TWIST NUMBER OF ALTERNATING KNOTS

2009 ◽  
Vol 18 (12) ◽  
pp. 1651-1662
Author(s):  
ROBERT G. TODD
Keyword(s):  
Exact Sequence ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Absolute Value ◽  
Kauffman Bracket ◽  
The Absolute ◽  
Twist Number ◽  
Skein Relation

In A Volumish Theorem for the Jones Polynomial, by O. Dasbach and X. S. Lin, it was shown that the sum of the absolute value of the second and penultimate coefficient of the Jones polynomial of an alternating knot is equal to the twist number of the knot. Here we give a new proof of this result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.

Remarks on Chebyshev polynomials, Fibonacci polynomials and Kauffman bracket skein modules

2018 ◽  
Vol 27 (07) ◽  
pp. 1841007
Author(s):  
Robert Owczarek
Keyword(s):  
Chebyshev Polynomials ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Matrix Representations ◽  
Fibonacci Polynomials ◽  
Spin Structures ◽  
Skein Modules ◽  
Kauffman Bracket ◽  
Special Cases ◽  
Counting Solutions

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS

1989 ◽  
Vol 04 (13) ◽  
pp. 3351-3373 ◽  
Author(s):  
MO-LIN GE ◽  
LU-YU WANG ◽  
KANG XUE ◽  
YONG-SHI WU
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Feynman Diagrams ◽  
Cpt Symmetry ◽  
Group Representations ◽  
Reidemeister Moves ◽  
S Matrix ◽  
Ambient Isotopy ◽  
Definition Of ◽  
Bracket Polynomials

By employing techniques familiar to particle physicists, we develop Kauffman’s state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman’s bracket polynomials. This procedure is explicitiy carried out for the N=3 and 4 cases. N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

scholarly journals THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF

2013 ◽  
Vol 88 (3) ◽  
pp. 407-422
Author(s):  
BOŠTJAN GABROVŠEK
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Nonorientable Surface ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

AbstractKhovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.

Mobility in Student Data Systems

2017 ◽  
Author(s):  
Ron Avi Astor ◽  
Linda Jacobson ◽  
Stephanie L. Wrabel ◽  
Rami Benbenishty ◽  
Diana Pineda
Keyword(s):  
Current Data ◽  
Data Systems ◽  
Homeless Students ◽  
New School ◽  
Student Data ◽  
Mobile Students ◽  
Definition Of ◽  
Additional Support ◽  
Changing Schools

For schools to be more proactive about addressing the needs of transitioning students and families, it’s important that district officials have a good sense of how often students are changing schools, who these students are, where they’re coming from, and where they’re going. Currently, there is wide variation in how states handle mobility in their student data systems. While some states have a specific definition of mobility, there are also differences in those definitions. By law, states track migrant and homeless students, but not all flag other groups of students that are likely to be mobile, such as military-connected students or those in foster care. Another complication is that when students move, schools do not mark the reason for the transition. Without knowing the reason for the change, all mobile students are lumped into one category— movers. But, as the previous chapter showed, the circumstances surrounding a move can affect students in different ways and have implications for how schools respond. If a move is proactive, for example, the family and the child may feel less stress and the student might feel more positive about the experience. If the change into a new school is reactive—caused perhaps by a difficult financial situation or leaving a negative situation at another school— the student and parents might feel more anxiety about the new school and need additional support and friendship during that time. Current data systems and the information they provide make it very difficult for researchers to separate the effect of the school move from the effect of the circumstances surrounding the move. These are important distinctions for educators to consider. Data systems do allow for researchers and practitioners to understand if a student moved during the summer or during the academic year. The timing of moves may be suggestive of the type of move a student is making; proactive moves may be more likely to occur in the summer months when learning will not be disrupted. Mid-year moves may have a proactive element, such as families moving for a better job, but they may also be reactive in nature, such as a loss of housing.

Petrophysical Rocks: Electrofacies and Lithofacies

2014 ◽  
Author(s):  
John H. Doveton
Keyword(s):  
Particle Size ◽  
Physical Properties ◽  
Sedimentary Rocks ◽  
Well Control ◽  
The Core ◽  
Lateral Extent ◽  
Petrophysical Logs ◽  
Definition Of

Many years ago, the classification of sedimentary rocks was largely descriptive and relied primarily on petrographic methods for composition and granulometry for particle size. The compositional aspect broadly matches the goals of the previous chapter in estimating mineral content from petrophysical logs. With the development of sedimentology, sedimentary rocks were now considered in terms of the depositional environment in which they originated. Uniformitarianism, the doctrine that the present is the key to the past, linked the formation of sediments in the modern day to their ancient lithified equivalents. Classification was now structured in terms of genesis and formalized in the concept of “facies.” A widely quoted definition of facies was given by Reading (1978) who stated, “A facies should ideally be a distinctive rock that forms under certain conditions of sedimentation reflecting a particular process or environment.” This concept identifies facies as process products which, when lithified in the subsurface, form genetic units that can be correlated with well control to establish the geological architecture of a field. The matching of facies with modern depositional analogs means that dimensional measures, such as shape and lateral extent, can be used to condition reasonable geomodels, particularly when well control is sparse or nonuniform. Most wells are logged rather than cored, so that the identification of facies in cores usually provides only a modicum of information to characterize the architecture of an entire field. Consequently, many studies have been made to predict lithofacies from log measurements in order to augment core observations in the development of a satisfactory geomodel that describes the structure of genetic layers across a field. The term “electrofacies” was introduced by Serra and Abbott (1980) as a way to characterize collective associations of log responses that are linked with geological attributes. They defined electrofacies to be “the set of log responses which characterizes a bed and permits it to be distinguished from the others.” Electrofacies are clearly determined by geology, because physical properties of rocks. The intent of electrofacies identification is generally to match them with lithofacies identified in the core or an outcrop.

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