Quantum knots and mosaics

Lomonaco, Samuel J.; Kauffman, Louis H.

doi:10.1007/s11128-008-0076-7

Cited by 57 publications

(30 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lomonaco and Kauffman [9,10] invented a mosaic system to give a precise and workable definition of quantum knots representing an actual physical quantum system. Oh et al have developed a state matrix argument for the knot mosaic enumeration in the papers [7,16].…”

Section: Stage 1: Conversion To Ivs Mosaicsmentioning

confidence: 99%

Enumerating independent vertex sets in grid graphs

Lee

2016

Linear Algebra and its Applications

View full text Add to dashboard Cite

Abstract. A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented.

show abstract

Section: Stage 1: Conversion To Ivs Mosaicsmentioning

confidence: 99%

Enumerating independent vertex sets in grid graphs

Lee

2016

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…In making quantum physical models, these equivalences should correspond to unitary transformations of an appropriate Hilbert space. Accordingly, we have formulated a model for quantum knots [60] that meets these requirements. A quantum knot system represents the "quantum embodiment" of a closed knotted physical piece of rope.…”

Section: Figure 8 -Borromean Ringsmentioning

confidence: 99%

Topological quantum information theory

Kauffman¹,

Lomonaco

2010

Proceedings of Symposia in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds. IntroductionThis paper describes relationships between quantum topology and quantum computing. It is a modified version of Chapter 14 of our book [18] and an expanded version of [58]. Quantum topology is, roughly speaking, that part of low-dimensional topology that interacts with statistical and quantum physics. Many invariants of knots, links and three dimensional manifolds have been born of this interaction, and the form of the invariants is closely related to the form of the computation of amplitudes in quantum mechanics. Consequently, it is fruitful to move back and forth between quantum topological methods and the techniques of quantum information theory.We sketch the background topology, discuss analogies (such as topological entanglement and quantum entanglement), show direct correspondences between certain topological operators (solutions to the Yang-Baxter equation) and universal quantum gates. We then describe the background for topological quantum computing in terms of Temperley-Lieb (we will sometimes abbreviate this to T L) recoupling theory. This is a recoupling theory that generalizes standard angular momentum recoupling theory, generalizes the Penrose theory of spin networks and is inherently topological. Temperley-Lieb recoupling Theory is based on the bracket polynomial model [37,44] for the Jones polynomial. It is built in terms of diagrammatic combinatorial topology. The same structure can be explained in terms of the SU (2) q quantum group, and has relationships with functional integration and Witten's approach to topological quantum field theory. Nevertheless, the approach given here will be unrelentingly elementary. Elementary, does not necessarily mean simple. In this case an architecture is built from simple beginnings and this archictecture and its recoupling language can be applied to many things including, e.g. colored Jones polynomials, Witten-Reshetikhin-Turaev invariants of three manifolds, topological quantum field theory and quantum computing.In quantum computing, the application of topology is most interesting...

show abstract

“…Wallace [Wa] has posted a method in which the barriers are drawn first, based on publications of G. Bain [Ba51] and I. Bain [Ba86], intended for graphic artists, by which anyone capable of following directions can handdraw Celtic knots, and which lends itself to implementation within a graphics system for creating computer-assisted art. Lomonaco and Kauffman [LK08] describe how a knot can be specified as a mosaic, in which the tiles contain crossings or the equivalent of barriers. Cromwell [Cr08] constructs Celtic knots by replacing crossings in basic plaitwork.…”

Section: Drawing a Celtic Knotmentioning

confidence: 99%

“…(For simplicity of exposition, we may sometimes say "knot" when our meaning is either a knot or a link.) We view Celtic representations of knots as a framework for organizing the study of knots, in the same spirit as, for example, knot mosaics [LK08], Gauss coding, or braid representations. Relevant background in knot theory is given, for example, by [Ad94], [Kau83], [Man04], and [Mu96].…”

Section: Introductionmentioning

confidence: 99%

A Celtic Framework for Knots and Links

Groß

Tucker

2010

Discrete Comput Geom

View full text Add to dashboard Cite

We describe a variant of a method used by modern graphic artists to design what are traditionally called Celtic knots, which are part of a larger family of designs called "mirror curves". It is easily proved that every such design specifies an alternating projection of a link. We use medial graphs and graph minors to prove, conversely, that every alternating projection of a link is topologically equivalent to some Celtic link, specifiable by this method. We view Celtic representations of knots as a framework for organizing the study of knots, rather like knot mosaics or braid representations. The formalism of Celtic design suggests some new geometric invariants of links and some new recursively specifiable sequences of links. It also leads us to explore new variations of problems regarding such sequences, including calculating formulae for infinite sequences of knot polynomials. This involves a confluence of ideas from knot theory, topological graph theory, and the theory of orthogonal graph drawings.

show abstract

Quantum knots and mosaics

Cited by 57 publications

References 19 publications

Enumerating independent vertex sets in grid graphs

Enumerating independent vertex sets in grid graphs

Topological quantum information theory

A Celtic Framework for Knots and Links

Contact Info

Product

Resources

About