scholarly journals A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 426 ◽  
Author(s):  
Stefan Heusler ◽  
Malte Ubben
Keyword(s):  
Hilbert Space ◽  
Braid Group ◽  
Knot Theory ◽  
Arbitrary Spin ◽  
Paper Strip ◽  
Spin States ◽  
Quantum Phases ◽  
Before And After ◽  
Jones Polynomials ◽  
Knots And Links

A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.

scholarly journals A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1399
Author(s):  
Stefan Heusler ◽  
Malte Ubben
Keyword(s):  
Hilbert Space ◽  
Physics Education ◽  
Particle Creation ◽  
Heegaard Splitting ◽  
Paper Strip ◽  
Torus Knots ◽  
Quantum Phases ◽  
Gauge Symmetries ◽  
Virtual Particles ◽  
The Status

The Heegaard splitting of S U ( 2 ) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S 1 → S 3, which can be related to ( 2 j , 2 ) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions ( I , J , K ), which are crucial to understand the Hopf mapping S 3 → S 2. We discuss the relationship between observables on the Bloch sphere S 2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a h a p t i c model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

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.

scholarly journals BAND SURGERY ON KNOTS AND LINKS

2010 ◽  
Vol 19 (12) ◽  
pp. 1535-1547 ◽  
Author(s):  
TAIZO KANENOBU
Keyword(s):  
Knot Theory ◽  
Site Specific ◽  
Site Specific Recombination ◽  
Knots And Links

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.

scholarly journals Virtual braids and virtual curve diagrams

2015 ◽  
Vol 24 (13) ◽  
pp. 1541001 ◽  
Author(s):  
Oleg Chterental
Keyword(s):  
Automorphism Group ◽  
Braid Group ◽  
Knot Theory ◽  
Class Group ◽  
Mapping Class ◽  
Virtual Knot ◽  
Injective Homomorphism ◽  
Virtual Links ◽  
Upper Half Plane ◽  
Virtual Knot Theory

There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].

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 Jones polynomials and classical conjectures in knot theory

Topology ◽  
1987 ◽  
Vol 26 (2) ◽  
pp. 187-194 ◽  
Author(s):  
Kunio Murasugi
Keyword(s):  
Knot Theory ◽  
Jones Polynomials

scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

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