scholarly journals Quantum automata, braid group and link polynomials

2007 ◽  
Vol 7 (5&6) ◽  
pp. 479-503
Author(s):  
S. Garnerone ◽  
A. Marzuoli ◽  
M. Rasetti
Keyword(s):  
Braid Group ◽  
Theoretical Background ◽  
Simons Theory ◽  
Tensor Algebra ◽  
Spin Network ◽  
Quantum Automata ◽  
Low Dimensional Topology ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Low Dimensional

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) $SU(2)$ Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link $L$ on $2N$ strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index $2N$, on the other. The growth rate of the time complexity function in terms of the integer $k$ appearing in the root of unity $q$ can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern--Simons theory.

Quantum Computation of Universal Link Invariants

2006 ◽  
Vol 13 (04) ◽  
pp. 373-382 ◽  
Author(s):  
Silvano Garnerone ◽  
Annalisa Marzuoli ◽  
Mario Rasetti
Keyword(s):  
Tensor Algebra ◽  
Network Simulator ◽  
Spin Network ◽  
Quantum Automata ◽  
Link Invariants ◽  
Root Of Unity ◽  
Braid Index ◽  
Finite State ◽  
Jones Polynomials ◽  
The One

In the framework of the spin-network simulator based on the SU q(2) tensor algebra, we implement families of finite state quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are coloured Jones polynomials. The automaton calculation of the polynomial of a link L on n strands at any fixed root of unity q is bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index n, on the other.

QFT, STRINGS AND LOW-DIMENSIONAL TOPOLOGY

2002 ◽  
Vol 17 (35) ◽  
pp. 2289-2295
Author(s):  
HU SEN
Keyword(s):  
Mirror Symmetry ◽  
Theoretical Physics ◽  
Yang Mills ◽  
Low Dimensional Topology ◽  
Witten Theory ◽  
Close Relationship ◽  
Jones Polynomials ◽  
Recent Developments ◽  
Low Dimensional ◽  
Chern Simons

In between the 80's and 90's we witnessed deep interactions between mathematics and theoretical physics, especially in the understanding of low-dimensional topology in terms of quantum field theory. For example, Jones polynomials (Chern–Simons–Witten theory), Donaldson and Seiberg–Witten invariants (SUSY Yang–Mills theory) and mirror symmetry (T duality in strings) are all naturally understood in terms of QFT and strings. Recent developments indicate a close relationship between gauge theory and gravity theory both in physics and in low-dimensional topology. We shall survey these developments and report some of our work. We shall also find that the keys to connect geometric and physical objects are through symmetry and quantization.

Quantum theories and geometric topology: A chapter in physical mathematics

2014 ◽  
Vol 11 (07) ◽  
pp. 1460024
Author(s):  
Kishore Marathe
Keyword(s):  
Topological String ◽  
Geometric Topology ◽  
Simons Theory ◽  
Quantum Field Theories ◽  
Quantum Theories ◽  
Low Dimensional Topology ◽  
Black Hole Radiation ◽  
Topological Quantum ◽  
Low Dimensional ◽  
String Amplitudes

In recent years, the interaction between geometric topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. We discuss some topics from low-dimensional topology where this has led to new viewpoints as well as new results. They include categorification of knot polynomials and a special case of the gauge theory to string theory correspondence in the Euclidean version of the theories, where exact results are available. We show how the Witten–Reshetikhin–Turaev invariant in SU (n) Chern–Simons theory on S3 is related via conifold transition to the all-genus generating function of the topological string amplitudes on a Calabi–Yau manifold. This result can be thought of as an interpretation of TQFT as topological quantum gravity (TQG). After a brief discussion of Perelman's work on the geometrization conjecture and its relation to gravity, we comment on some recent work on black hole radiation and its relation to mock moonshine.

scholarly journals AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

2008 ◽  
Vol 06 (supp01) ◽  
pp. 773-778 ◽  
Author(s):  
MARIO RASETTI ◽  
SILVANO GARNERONE ◽  
ANNALISA MARZUOLI
Keyword(s):  
Jones Polynomial ◽  
Quantum Algorithm ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
Oriented Link ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. The construction exploits the q-deformed spin network as computational background. The complexity of such algorithm is bounded above linearly by the number of crossings of the link, and polynomially by the number of link strands.

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

Low-Dimensional Topology and Number Theory

2010 ◽  
pp. 2101-2163 ◽  
Author(s):  
Paul Gunnells ◽  
Walter Neumann ◽  
Adam Sikora ◽  
Don Zagier
Keyword(s):  
Number Theory ◽  
Low Dimensional Topology ◽  
Low Dimensional

scholarly journals Cross-sections of unknotted ribbon disks and algebraic curves

2019 ◽  
Vol 155 (2) ◽  
pp. 413-423
Author(s):  
Kyle Hayden
Keyword(s):  
Cross Sections ◽  
Algebraic Curves ◽  
American Mathematical Society ◽  
Geometric Topology ◽  
Mathematical Society ◽  
Advanced Mathematics ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Holomorphic Disks

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

scholarly journals Algebraic Structures in Low-Dimensional Topology

2014 ◽  
Vol 11 (2) ◽  
pp. 1403-1458 ◽  
Author(s):  
Louis Kauffman ◽  
Vassily Manturov ◽  
Kent Orr ◽  
Robert Schneiderman
Keyword(s):  
Algebraic Structures ◽  
Low Dimensional Topology ◽  
Low Dimensional
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