A pr 2 00 4 Quantum Knots (original) (raw)
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This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and examples are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a separate, network model for quantum evolution and measurement, where
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Series on Knots and Everything, 2011
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots, This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is Intended to represent the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial, configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some Interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot Is simply the orbit of the quantum knot under the action of the ambient group. We Investigate quantum observables which are Invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and brieiy look at the quantum tunneling of overcrossings into endercrossings. A basic building block In this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.
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Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do …
Arxiv preprint arXiv:0910.5891
Abstract. In the paper ”Quantum knots and mosaics,” [25], the definition of quantum knots was based on the planar projections of knots (ie, on knot diagrams) and the Reidemeister moves on these projections. In this paper, we take a different tack by creating a definition of ...
Quantum knots and lattices, or a blueprint for quantum systems that do rope tricks
Proceedings of Symposia in Applied Mathematics, 2010
In the paper "Quantum knots and mosaics," [25], the definition of quantum knots was based on the planar projections of knots (i.e., on knot diagrams) and the Reidemeister moves on these projections. In this paper, we take a different tack by creating a definition of quantum knots based on the cubic honeycomb decomposition of 3-space R 3 (i.e., the cubic tesselation L ℓ of R 3 consisting of 2 −ℓ × 2 −ℓ × 2 −ℓ cubes) and a new set of knot moves, called wiggle, wag, and tug, which unlike the two dimensional Reidemeister moves are truly three dimensional moves. These new moves have been so named because they mimic how a dog might wag its tail. We believe that these two different approaches to defining quantum knots are essentially equivalent, but that the above three dimensional moves have a definite advantage when it comes to the applications of knot theory to physics. More specifically, we contend that the new moves wiggle, wag, and tug are more "physics-friendly" than the Reidemeister ones. For unlike the Reidemeister moves, the new moves are three dimensional moves that respect the differential geometry of 3-space, which is indeed an essential component of physics. And moreover, unlike the Reidemeister moves, they can be transformed into infinitesimal moves and differential forms, which structures can be seamlessly interwoven with the equations of physics. Our basic building block for constructing a quantum knot is a lattice knot, which is a knot in 3-space constructed from the edges of the cubic honeycomb L ℓ. We then create a Hilbert space by identifying each edge of a bounded n × n × n region of the cubic honeycomb with a qubit. Lattice knots within this region then form the basis of a sub-Hilbert space K (ℓ,n). The states of K (ℓ,n) are called quantum knots. The knot moves, wiggle, wag, and tug, are then naturally identified with the generators of a unitary group Λ ℓ,n , called the lattice ambient group, acting on the Hilbert space K (ℓ,n). This definition of a quantum knot can be viewed as a blueprint for the construction of an actual physical quantum system that represents the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this quantum knot system, represents the state of such a knotted closed piece of rope, i.e., the particular spacial configuration of the knot tied in the rope. The lattice ambient group Λ ℓ,n represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. After defining quantum knot type, we investigate quantum observables which are invariants of quantum knot type. Moreover, we also study the Hamiltonians associated with the generators of the lattice ambient group.
A chapter in physical mathematics: theory of knots in the sciences
2000
A systematic study of knots was begunin the second half of the 19th century by Tait and his followers. They were motivated by Kelvin's theory of atoms modelled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could beexpressed in terms of properties of knots such as the knot invariants. Even though Kelvin's theory did not work, the theory of knots grew as a sub eld of combinatorial topology. Recently new invariants of knots have been discovered and they have led to the solution of long standing problems in knot theory. Surprising connections between the theory of knots and statistical mechanics, quantum groups and quantum eld theory are emerging. We give a geometric formulation of some of these invariants using ideas from topological quantum eld theory. We also discuss some recent connections and application of knot theory to problems in Physics, Chemistry and Biology. It is interesting to note that as we stand on the threshold of the new millenium, di cult questions arising in the sciences continue to serve as a driving force for the development of new mathematical tools needed to understand and answer them.
Topological aspects of quantum entanglement
Quantum Information Processing
explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang-Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang-Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang-Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU (2) representation of the three-strand braid group that models the Jones polynomial for closures of threestrand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
The Knot Theory A Comprehensive Discussion
Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-dimensional space. By examining how these loops can be twisted, entangled, and deformed without cutting or breaking, knot theory provides profound insights into the structure of our universe, ranging from the microscopic strands of DNA to the complex topology of quantum fields. With its roots in 19th-century mathematics and its influence extending to modern physics, biology, and computer science, knot theory bridges pure mathematical abstraction and practical applications. This paper offers a comprehensive exploration of knot theory, starting with its foundational concepts and definitions. It examines the tools mathematicians use to classify knots, including powerful invariants like the Jones polynomial and knot groups. The paper also highlights advanced topics, such as the relationship between braids and knots and the role of knot theory in higher dimensions. Finally, it sheds light on the practical relevance of knot theory, showcasing its applications in understanding DNA replication, quantum computing, and fluid dynamics. By weaving together intuition, rigorous mathematics, and real-world applications, this paper aims to provide both an introduction to knot theory and a glimpse into the open problems that continue to challenge and inspire mathematicians. Whether you're a mathematician, a scientist, or simply a curious mind, this journey through the tangled world of knots promises to be as intriguing as the knots themselves.
2011
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms for the Jones polynomial and relations with Khovanov homology. The purpose of this paper is to place such constructions in a general context of the quantization of mathematical structures.