arXiv subjects

Louis H. Kauffman

Publications and source records attributed to Louis H. Kauffman.

At least 145 records · Page 8Linked to original sources

arXiv2007

On the minimum number of colors for knots

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva Transformations. Each of these Transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds on these minima. We conjecture on the value of these minima. We apply these Transformations to rational knots.

math.GT↗

arXiv2007

Spin Networks and Anyonic Topological Computing II

We review the q-deformed spin network approach to topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a shape suitable for computation and algebraic work.

quant-ph↗

arXiv2007

A 3-Stranded Quantum Algorithm for the Jones Polynomial

Let K be a 3-stranded knot (or link), and let L denote the number of crossings in K. Let $ε_{1}$ and $ε_{2}$ be two positive real numbers such that $ε_{2}$ is less than or equal to 1. In this paper, we create two algorithms for computing the value of the Jones polynomial of K at all points $t=exp(iϕ)$ of the unit circle in the complex plane such that the absolute value of $ϕ$ is less than or equal to $π/3$. The first algorithm, called the classical 3-stranded braid (3-SB) algorithm, is a classical deterministic algorithm that has time complexity O(L). The second, called the quantum 3-SB algorithm, is a quantum algorithm that computes an estimate of the Jones polynomial of K at $exp(iϕ))$ within a precision of $ε_{1}$ with a probability of success bounded below by $1-ε_{2}%. The execution time complexity of this algorithm is O(nL), where n is the ceiling function of (ln(4/ε_{2}))/(2(ε_{2})^2). The compilation time complexity, i.e., an asymptotic measure of the amount of time to assemble the hardware that executes the algorithm, is O(L).

quant-ph↗

arXiv2007

Topological-Like Features in Diagrammatical Quantum Circuits

In this paper, we revisit topological-like features in the extended Temperley--Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We perform these quantum circuits and derive characteristic equations for them with the help of topological-like operations. Furthermore, we comment on known diagrammatical approaches to quantum information phenomena from the perspectives of both tensor categories and topological quantum field theories. Moreover, we remark on the proposal for categorical quantum physics and information to be described by dagger ribbon categories.

quant-ph↗

arXiv2007

Biquandles for Virtual Knots

This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms are associated to oriented classical and virtual crossings in the diagram.

math.GT↗

arXiv2007

$q$ - Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation

We review the q-deformed spin network approach to 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.

quant-ph↗

arXiv2006

The L-Move and Virtual Braids

This paper is a short introduction to and statement of the main theorems of our paper "Virtual Braids and the L-Move", JKTR, Vol. 15, No. 6 (2006), pp. 773-811. See also arxiv:Math.GT/0507035.

math.GT↗

arXiv2006

Virtual Extension of Temperley--Lieb Algebra

The virtual knot theory is a new interesting subject in the recent study of low dimensional topology. In this paper, we explore the algebraic structure underlying the virtual braid group and call it the virtual Temperley--Lieb algebra which is an extension of the Temperley--Lieb algebra by adding the group algebra of the symmetrical group. We make a connection clear between the Brauer algebra and virtual Temperley--Lieb algebra, and show the algebra generated by permutation and its partial transpose to be an example for the virtual Temperley--Lieb algebra and its important quotients.

math-ph↗

arXiv2006

Lune -- Free Knot Graphs

This paper is an exploration of simple four-regular graphs in the plane (i.e. loopless and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921 -- 2005) whose fascination with graphs of knots inspired this work and with whom we had the pleasure of developing this paper. We prove that for v (the number of nodes) greater than or equal to 8 there always exist such knot-graphs. Our proof gives rise to a class of "tight" graphs that do not submit to the recursive procedure that generates graphs with v+2 nodes from "admissible" graphs with v nodes. We then classify such tight graphs.

math.CO↗

arXiv2006

Quantum Hidden Subgroup Algorithms: An Algorithmic Toolkit

One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor's quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an "alphabetic building blocks approach," we use these primitives to form an "algorithmic toolkit" for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover's algorithm is most surprisingly "almost" a QHS algorithm, and how this result suggests the possibility of an even more complete "algorithmic tookit" beyond the QHS algorithms.

quant-ph↗

arXiv2006

Virtual Braids and the L--Move

In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove the Virtual Markov Theorem. One benefit of this approach is a fully local algebraic formulation of the Theorem.

math.GT↗

arXiv2006

Permutation and Its Partial Transpose

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang--Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the "PPT" algebra which guides the construction of multipartite symmetric states. The virtual knot theory having permutation as a virtual crossing provides a topological language describing quantum computation having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley--Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang--Baxter equations; and virtual Temperley--Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the "ABPK" diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies nontrivial unitary braid representations with universal quantum gates, and derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

quant-ph↗

arXiv2006

Remarks on Formal Knot Theory

This paper is an introduction to the state sum model for the Alexander-Conway polynomial that was introduced in the the author's book "Formal Knot Theory" (Princeton University Press, 1983). The article outlines how Alexander's original definition of the polynomial as the determinant of a matrix associated with the link diagram can be reformulated as a state summation over combinatorial configurations of the diagram. Facts about the state model, the Clock Theorem (that makes this model tick) and relations with other state models such as the bracket polynomial, are discussed. The paper includes a miniature introduction to Khovanov homology. This paper will appear in the forthcoming Dover republication of Formal Knot Theory.

math.GT↗

arXiv2006

Virtual Knot Theory

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss codes via knot diagrams with virtual crossings. This paper gives basic results and examples (such as non-trivial virtual knots with trivial Jones polynomial), studies fundamental group and quandles of virtual knots, extensions of the bracket and Jones polynomials, quantum link invariants with virtual framings, Vassiliev invariants and applications to knots in thickened surfaces.

math.GT↗

arXiv2006

Topological Quantum Computing and the Jones Polynomial

In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form exp(2$π$i/k). This description is given with two objectives in mind. The first is to describe the algorithm in such a way as to make explicit the underlying and inherent control structure. The second is to make this algorithm accessible to a larger audience.

quant-ph↗

arXiv2006

Spin Networks and Anyonic Topological Computing

We review the q-deformed spin network approact to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods produce a concise proof that quantum computation can be performed within a single representation of the Artin Braid Group.

quant-ph↗

arXiv2006

Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm ?

The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm.

quant-ph↗

arXiv2005

Universal Quantum Gate, Yang--Baxterization and Hamiltonian

It is fundamental to view unitary braiding operators describing topological entanglements as universal quantum gates for quantum computation. This paper derives a unitary solution of the Quantum Yang--Baxter equation via Yang--Baxterization and constructs the Hamiltonian responsible for the time-evolution of the unitary braiding operator.

quant-ph↗