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 Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

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.

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>

Image Representation, Filtering, and Natural Computing in a Multivalued Quantum System

2016 ◽  
pp. 28-56 ◽  
Author(s):  
Sanjay Chakraborty ◽  
Lopamudra Dey
Keyword(s):  
Image Processing ◽  
Quantum System ◽  
Quantum Physics ◽  
Image Representation ◽  
Quantum Image Processing ◽  
Natural Computing ◽  
Quantum Image ◽  
Image Pixels ◽  
Classical Image ◽  
And Storage

Image processing on quantum platform is a hot topic for researchers now a day. Inspired from the idea of quantum physics, researchers are trying to shift their focus from classical image processing towards quantum image processing. Storing and representation of images in a binary and ternary quantum system is always one of the major issues in quantum image processing. This chapter mainly deals with several issues regarding various types of image representation and storage techniques in a binary as well as ternary quantum system. How image pixels can be organized and retrieved based on their positions and intensity values in 2-states and 3-states quantum systems is explained here in detail. Beside that it also deals with the topic that focuses on the clear filteration of images in quantum system to remove unwanted noises. This chapter also deals with those important applications (like Quantum image compression, Quantum edge detection, Quantum Histogram etc.) where quantum image processing associated with some of the natural computing techniques (like AI, ANN, ACO, etc.).

scholarly journals Are living beings extended autopoietic systems? An embodied reply

2019 ◽  
Vol 28 (1) ◽  
pp. 3-13 ◽  
Author(s):  
Mario Villalobos ◽  
Pablo Razeto-Barry
Keyword(s):  
Research Program ◽  
Original Formulation ◽  
Autopoietic Systems ◽  
Form Part ◽  
Definition Of ◽  
Autopoietic Theory ◽  
Extended Interpretation

Building on the original formulation of the autopoietic theory (AT), extended enactivism argues that living beings are autopoietic systems that extend beyond the spatial boundaries of the organism. In this article, we argue that extended enactivism, despite having some basis in AT’s original formulation, mistakes AT’s definition of living beings as autopoietic entities. We offer, as a reply to this interpretation, a more embodied reformulation of autopoiesis, which we think is necessary to counterbalance the (excessively) disembodied spirit of AT’s original formulation. The article aims to clarify and correct what we take to be a misinterpretation of AT as a research program. AT, contrary to what some enactivists seem to believe, did not (and does not) intend to motivate an extended conception of living beings. AT’s primary purpose, we argue, was (and is) to provide a universal individuation criterion for living beings, these understood as discrete bodies that are embedded in, but not constituted by, the environment that surrounds them. However, by giving a more explicitly embodied definition of living beings, AT can rectify and accommodate, so we argue, the enactive extended interpretation of autopoiesis, showing that although living beings do not extend beyond their boundaries as autopoietic unities, they do form part, in normal conditions, of broader autopoietic systems that include the environment.

K-Theory and the Jones Polynomial

2018 ◽  
Vol 30 (06) ◽  
pp. 1840002
Author(s):  
Michael F. Atiyah ◽  
Carlos Zapata-Carratala
Keyword(s):  
Jones Polynomial ◽  
New Approach ◽  
K Theory ◽  
Definition Of

In this paper, we present a new approach to the definition of the Jones polynomial using equivariant K-theory. Dedicated to Ludwig Faddeev

scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

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