scholarly journals ON THE DEFINITION OF GRAPH INDEX

2013 ◽  
Vol 94 (3) ◽  
pp. 417-429 ◽  
Author(s):  
A. STOIMENOW
Keyword(s):  
Knot Theory ◽  
Braid Index ◽  
Definition Of

AbstractThis paper discusses a flaw in Murasugi–Przytycki’s Memoir ‘An index of a graph with applications to knot theory’ [Mem. Amer. Math. Soc. 106 (1993)]. We point out and partly fix a gap occurring in the proof of Murasugi–Przytycki’s braid index inequalities involving the graph index. We explain why their notion of index fails to precisely reflect the reduction of Seifert circles by their diagram move, and redefine the index to account for that discrepancy.

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 A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

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>

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.

A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS

1999 ◽  
Vol 11 (07) ◽  
pp. 929-945 ◽  
Author(s):  
J. JUYUMAYA
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Knot Theory ◽  
Hecke Algebra ◽  
Definition Of

In this work we define a new algebra. The definition of our algebra arises naturally in the study of certain generators (non standard) for Yokonuma–Hecke algebra [8]. This algebra is linked to the Knot theory via the Vassiliev algebra defined by J. Baez [2].

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 Space of chord diagrams on spherical curves

2019 ◽  
Vol 30 (12) ◽  
pp. 1950060 ◽  
Author(s):  
Noboru Ito
Keyword(s):  
Knot Theory ◽  
Reidemeister Move ◽  
Plane Curves ◽  
Type I ◽  
Main Result Theorem ◽  
Chord Diagrams ◽  
Result Theorem ◽  
Ambient Isotopy ◽  
Definition Of

In this paper, we give a definition of [Formula: see text]-valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by [Formula: see text]. Then, we introduce certain elements of the free [Formula: see text]-module generated by the chord diagrams with at most [Formula: see text] chords, called relators of Type (I) ((SI[Formula: see text]I), (WI[Formula: see text]I), (SI[Formula: see text]I[Formula: see text]I), or (WI[Formula: see text]I[Formula: see text]I), respectively), and introduce another function [Formula: see text] derived from [Formula: see text]. The main result (Theorem 1) shows that if [Formula: see text] vanishes for the relators of Type (I) ((SI[Formula: see text]I), (WI[Formula: see text]I), (SI[Formula: see text]I[Formula: see text]I), or (WI[Formula: see text]I[Formula: see text]I), respectively), then [Formula: see text] is invariant under the Reidemeister move of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I, or weak RI[Formula: see text]I[Formula: see text]I, respectively) that is defined in [N. Ito and Y. Takimura, [Formula: see text] and weak [Formula: see text] homotopies on knot projections, J. Knot Theory Ramifications 22 (2013) 1350085 14 pp].

On braids and groups Gnk

2015 ◽  
Vol 24 (13) ◽  
pp. 1541009 ◽  
Author(s):  
Vassily Olegovich Manturov ◽  
Igor Mikhailovich Nikonov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Braid Groups ◽  
The Other ◽  
Pure Braid Group ◽  
And Topology ◽  
Other Hand ◽  
Definition Of

In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of [Formula: see text]-free braid groups [Formula: see text]. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to [Formula: see text]-free braid groups for important cases [Formula: see text]. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

Introductory paper

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.

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

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.

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