A homological definition of the Jones polynomial

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.

Download Full-text

The Khovanov homology of knots

10.26686/wgtn.17013602 ◽

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>

Download Full-text

THE JONES POLYNOMIAL AND THE INTERSECTION NUMBERS OF TWISTED CYCLES ASSOCIATED WITH A SELBERG TYPE INTEGRAL

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511008887 ◽

2011 ◽

Vol 20 (03) ◽

pp. 469-496 ◽

Cited By ~ 2

Author(s):

KATSUHISA MIMACHI

Keyword(s):

Special Functions ◽

Jones Polynomial ◽

Intersection Number ◽

Homology Theory ◽

Intersection Numbers ◽

Type Integral ◽

Jones Polynomials ◽

Twisted Homology ◽

Hypergeometric Type ◽

Definition Of

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

Download Full-text

AN ORIENTED MODEL FOR KHOVANOV HOMOLOGY

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216510007863 ◽

2010 ◽

Vol 19 (02) ◽

pp. 291-312 ◽

Cited By ~ 8

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.

Download Full-text

AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS

International Journal of Modern Physics A ◽

10.1142/s0217751x89001370 ◽

1989 ◽

Vol 04 (13) ◽

pp. 3351-3373 ◽

Cited By ~ 21

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.

Download Full-text

The Khovanov homology of knots

10.26686/wgtn.17013602.v1 ◽

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>

Download Full-text

Introductory paper

Symposium - International Astronomical Union ◽

10.1017/s0074180900053031 ◽

1966 ◽

Vol 24 ◽

pp. 3-5

Author(s):

W. W. Morgan

Keyword(s):

Line Intensity ◽

Classification Scheme ◽

Two Dimensional ◽

Spectral Classification ◽

Dimensional Array ◽

Rr Lyrae ◽

Metallic Line ◽

Introductory Paper ◽

Parameter Varying ◽

Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

Download Full-text

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

International Astronomical Union Colloquium ◽

10.1017/s0252921100050867 ◽

1975 ◽

Vol 26 ◽

pp. 21-26

Keyword(s):

Coordinate System ◽

Working Group ◽

General Character ◽

Coordinate Systems ◽

Reference Coordinate System ◽

Group 2 ◽

Definition Of ◽

Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

Download Full-text

Symbiotic Stars at Optical, Infrared and Radio Wavelengths

International Astronomical Union Colloquium ◽

10.1017/s025292110007336x ◽

1979 ◽

Vol 46 ◽

pp. 125-149 ◽

Cited By ~ 3

Author(s):

David A. Allen

Keyword(s):

Temperature Regime ◽

Type Star ◽

Stellar Systems ◽

Symbiotic Stars ◽

Lower Excitation ◽

Definition Of ◽

Original Type

No paper of this nature should begin without a definition of symbiotic stars. It was Paul Merrill who, borrowing on his botanical background, coined the termsymbioticto describe apparently single stellar systems which combine the TiO absorption of M giants (temperature regime ≲ 3500 K) with He II emission (temperature regime ≳ 100,000 K). He and Milton Humason had in 1932 first drawn attention to three such stars: AX Per, CI Cyg and RW Hya. At the conclusion of the Mount Wilson Ha emission survey nearly a dozen had been identified, and Z And had become their type star. The numbers slowly grew, as much because the definition widened to include lower-excitation specimens as because new examples of the original type were found. In 1970 Wackerling listed 30; this was the last compendium of symbiotic stars published.

Download Full-text