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 $m \! \times \! n$ 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

Virtual Mosaic Knots

The study of knots and links is a main focus of the mathematical branch of topology. Classical knot theory studies knots embedded in 3-dimensional real space and has been a primary field of study since the 1960’s. Virtual knot theory, first introduced by Kauffman in 1999, studies knots embedded in thickened surfaces. Lomanoco and Kauffman introduced mosaic diagrams in order to build a quantum knot system in 2008. In 2009, Garduño extended these mosaic diagrams to include virtual knots. In order to represent knots on surfaces, Ganzell and Henrich introduced virtual mosaic knot theory in 2020 by placing knots onto × polygonal representations of surfaces. We extend the idea of virtual mosaic knot theory to include virtual rectangular mosaics, a placement of virtual knots onto × polygonal representations of surfaces, as well as row mosaics, a placement of virtual knots onto 1 × polygonal representations of surfaces. In this thesis, we introduce virtual rectangular mosaics and give two rectangular mosaic invariants called the tile number and row number. Included as an appendix, we give a complete row mosaic tabulation of knots with 8 or fewer crossings and virtual knots up to 4 crossings

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

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