37 research outputs found
Quantum Knots and Mosaics
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
Period and toroidal knot mosaics
Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on
`Quantum knots and mosaics' to give a precise and workable definition of
quantum knots, intended to represent an actual physical quantum system. A knot
(m,n)-mosaic is an matrix whose entries are eleven mosaic
tiles, representing a knot or a link by adjoining properly. In this paper we
introduce two variants of knot mosaics: period knot mosaics and toroidal knot
mosaics, which are common features in physics and mathematics. We present an
algorithm producing the exact enumeration of period knot (m,n)-mosaics for any
positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m
and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We
also analyze the asymptotics of the growth rates of their cardinality
Tile Number and Space-Efficient Knot Mosaics
In this paper we introduce the concept of a space-efficient knot mosaic. That
is, we seek to determine how to create knot mosaics using the least number of
non-blank tiles necessary to depict the knot. This least number is called the
tile number of the knot. We determine strict bounds for the tile number of a
knot in terms of the mosaic number of the knot. In particular, if is the
tile number of a prime knot with mosaic number , then if is even and if is odd. We also
determine the tile number of several knots and provide space-efficient knot
mosaics for each of them.Comment: The original version of this article was split into two articles
during refereein
On Upper Bounds for Toroidal Mosaic Numbers
In this paper, we work to construct mosaic representations of knots on the
torus, rather than in the plane. This consists of a particular choice of the
ambient group, as well as different definitions of contiguous and suitably
connected. We present conditions under which mosaic numbers might decrease by
this projection, and present a tool to measure this reduction. We show that the
order of edge identification in construction of the torus sometimes yields
different resultant knots from a given mosaic when reversed. Additionally, in
the Appendix we give the catalog of all 2 by 2 torus mosaics.Comment: 10 pages, 111 figure