The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot [Formula: see text]-mosaic is an [Formula: see text] array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number [Formula: see text] of knot [Formula: see text]-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant [Formula: see text] and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: [Formula: see text]
The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot [Formula: see text]-mosaic is an [Formula: see text] array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number [Formula: see text] of knot [Formula: see text]-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant [Formula: see text] and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: [Formula: see text]
Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix of mosaic tiles (T0 through T10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. D(m,n) is the total number of all knot (m,n)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. D(m,n) is already found for m,n≤6 by the authors. In this paper, we construct an algorithm producing the precise value of D(m,n) for m,n≥2 that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. D(m,n)=2∥(Xm−2+Om−2)n−2∥ where 2m−2×2m−2 matrices Xm−2 and Om−2 are defined by $$ X_{k+1} = \begin{bmatrix} X_k & O_k \\ O_k & X_k \end{bmatrix} \ \mbox{and } \ O_{k+1} = \begin{bmatrix} O_k & X_k \\ X_k & 4 \, O_k \end{bmatrix} $$ for k=0,1,⋯,m−3, with 1×1 matrices X0=[1] and O0=[1]. Here ∥N∥ denotes the sum of all entries of a matrix N. For n=2, (Xm−2+Om−2)0 means the identity matrix of size 2m−2×2m−2.
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 briefly look at the quantum tunneling of overcrossings into undercrossings. 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.
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of “knot invariants,” among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a “universal problem,” namely, the hardest problem that a quantum computer can efficiently handle.
In 2001, Michael Berry4 published the paper ”Knotted Zeros in the Quantum States of Hydrogen” in Foundations of Physics. In this paper we show how to place Berry’s discovery in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space R3 to the complex plane and such that the inverse image of 0 in the complex plane contains a knotted curve in R3. We show that for knots in R3 this is a generic situation in that every smooth knot K in R3 has a smooth classifying map f : R3 −→ C (the complex plane) such that f−1(0) = K. This leaves open the question of characterizing just when such f are wave-functions for quantum systems. One can compare this result with the work of Mark Dennis and his collaborators and with the work of Lee Rudolph. Our approach provides great generality to the structure of knotted zeros of a wavefunction and opens up many new avenues for research in the relationships of quantum theory and knot theory. We show how this classifying construction can be related our previous work on two dimensional and three dimensional mosaic and lattice quantum knots.
FINDING: The Jones polynomial is a knot invariant constructible from quantum R-matrices (braid group representations), and its evaluation at roots of unity yields the computational resource for topological quantum computation — the same algebraic structure underlies both knot theory and TQFT. | MATH: Jones polynomial V_K(t) ∈ ℤ[t^{±1/2}]; skein relation: t⁻¹V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀). Braid group representation: ρ: B_n → GL(V^⊗n) via R-matrix satisfying Yang-Baxter equation (R⊗I)(I⊗R)(R⊗I) = (I⊗R)(R⊗I)(I⊗R). At t = e^{2πi/k}, the representation factors through Temperley-Lieb algebra TL_n(q) with q + q⁻¹ = −(t^{1/2} + t^{−1/2}). Quantum dimension d = −(q + q⁻¹) = [2]_q. Modular tensor category (MTC) data: fusion rules N_{ij}^k, S-matrix, T-matrix (topological spin), satisfying (ST)³ = e^{2πic/8}·S² (modularity). | CONNECTION: Root-of-unity evaluation t = e^{2πi/k} connects to: (1) quantum group U_q(sl₂) at q = e^{iπ/k} — deformation parameter is a root of unity, truncati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The study of anyonic systems as computational means has led to the exciting discovery of a new quantum algorithm. This algorithm provides a novel paradigm that fundamentally differs from searching (Grover, 1996) and factoring (Shor, 1997) algorithms. It is based on the particular behaviour of anyons and its goal is to evaluate Jones polynomials (Jones, 1985, 2005). These polynomials are topological invariants of knots and links, i.e., they depend on the global characteristics of their strands and not on their local geometry. The Jones polynomials were first connected to topological quantum field theories by Edward Witten (1989). Since then they have found applications in various areas of research, such as biology for DNA reconstruction (Nechaev, 1996) and statistical physics (Kauffman, 1991). The best known classical algorithm for the exact evaluation of the Jones polynomial demands exponential resources (Jaeger et al. , 1990). Employing anyons involves only a polynomial number of resources to produce an approximate answer to this problem (Freedman et al. , 2003b). Evaluating Jones polynomials by manipulating anyons resembles an analogue computer. Indeed, the idea is equivalent to the classical setup, where a wire is wrapped several times around a solenoid that confines magnetic flux. By measuring the current that runs through the wire one can obtain the number of times the wire was wrapped around the solenoid, i.e., their linking number. Similarly, by creating anyons and spanning links with their worldlines we are able to extract the Jones polynomials of these links (Kauffman and Lomanaco, 2006).