Papers by Louis H Kauffman
arXiv: Geometric Topology, 2020
The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an... more The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we define and prove a Khovanov homology theory for unoriented virtual links. (Virtual links include all classical links.) The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual (oriented) Jones polynomial: for classical or even links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new, simpler algorithm for computing integral Khovanov homology is described. Finally, we define unoriented Lee homology theory for virtual links based upon the unorien...
Journal of Social and Biological Systems, 1980
... 1 Form dynamics 173 These is nothing obscure or esoteric about our notational system. ... It ... more ... 1 Form dynamics 173 These is nothing obscure or esoteric about our notational system. ... It has often been pointed out that the formal structure of G. Spencer Brown's work, and a fortiori of ... expression grows in the pattern: I, II, iI, Ill, Viewed in space we would see something like iI f ...
ABSTRACT This paper has been withdrawn because there is a fundamental error in the computations; ... more ABSTRACT This paper has been withdrawn because there is a fundamental error in the computations; with the right computational scheme it seems to be just a version of the Jones polynomial Comment: This paper has been withdrawn because there is a fundamental error in the computations; with the right computational scheme it seems to be just a version of the Jones polynomial
Logic and Algebraic Structures in Quantum Computing
In this paper we review the topological model for the quaternions based upon the Dirac string tri... more In this paper we review the topological model for the quaternions based upon the Dirac string trick. We then extend this model, to create a model for the octonions - the non-associative generalization of the quaternions.
This paper discusses ER = EPR. Given a background space and a quantum tensor network, we describe... more This paper discusses ER = EPR. Given a background space and a quantum tensor network, we describe how to construct a new topological space, that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
Symmetry, 2022
This paper shows how gauge theoretic structures arise in a noncommutative calculus where the deri... more This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton’s equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper.
In this manuscript, we introduce a method to measure entanglement of curves in 3-space that exten... more In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.
Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the ... more Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's mu and bar mu invariants to welded and virtual links. We conclude this paper with several examples, and compute the mu invariants using the Magnus expansion and Polyak's skein relation for the mu invariants.
In this paper we review the topological model for the quaternions based upon the Dirac string tri... more In this paper we review the topological model for the quaternions based upon the Dirac string trick. We then extend this model, to create a model for the octonions - the non-associative generalization of the quaternions.
In this paper we explore the boundary between biology and the study of formal systems (logic). In... more In this paper we explore the boundary between biology and the study of formal systems (logic). In the end, we arrive at a summary formalism, a chapter in "boundary mathematics" where there are not only containers <> but also extainers ><, entities open to interaction and distinguishing the space that they are not. The boundary algebra of containers and extainers is to biologic what boolean algebra is to classical logic. We show how this formalism encompasses significant parts of the logic of DNA replication, the Dirac formalism for quantum mechanics, formalisms for protein folding and the basic structure of the Temperley Lieb algebra at the foundations of topological invariants of knots and links.
A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknott... more A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surprisingly difficult, since it has been shown that knot diagrams may need to be made more complicated before they may be simplified. We do not yet know, however, how much more complicated they must get. We give an introduction to the work of Dynnikov who discovered the key use of arc--presentations to solve the problem of finding a way to detect the unknot directly from a diagram of the knot. Using Dynnikov's work, we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves. We also apply Dynnikov's results to find an upper bound for the number of moves required in an unknotting sequence.
We present chemlambda (or the chemical concrete machine), an artificial chemistry with the follow... more We present chemlambda (or the chemical concrete machine), an artificial chemistry with the following properties: (a) is Turing complete, (b) has a model of decentralized, distributed computing associated to it, (c) works at the level of individ-ual (artificial) molecules, subject of reversible, but otherwise deterministic interactions with a small number of enzymes, (d) encodes information in the geometrical structure of the molecules and not in their numbers, (e) all interactions are purely local in space and time. This is part of a larger project to create computing, artificial chemistry and artificial life in a distributed context, using topological and graphical lan-guages.
This file contains the proofs to Theorems 4.1, 4.2 and 4.3 and Examples of knotoids
Journal of Knot Theory and Its Ramifications, 2021
We use the terms, knot product and local-move, as defined in the text of this paper. Let [Formula... more We use the terms, knot product and local-move, as defined in the text of this paper. Let [Formula: see text] be an integer [Formula: see text]. Let [Formula: see text] be the set of simple spherical [Formula: see text]-knots in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove that the map [Formula: see text] is bijective, where [Formula: see text]Hopf, and Hopf denotes the Hopf link. Let [Formula: see text] and [Formula: see text] be 1-links in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single pass-move, which is a local-move on 1-links. Let [Formula: see text] be a positive integer. Let [Formula: see text] denote the knot product [Formula: see text]. We prove the following: The [Formula: see text]-dimensional submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-submani...
Linear Algebra and its Applications, 2021
International Journal of Modern Physics A, 2018
In this paper we study unitary braid group representations associated with Majorana Fermions. Maj... more In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this result with the Ivanov braiding associated with Majorana operators. The paper generalizes observations of Kauffman and Lomonaco and of Mo-Lin Ge to show that certain strings of Majorana operators give rise to extraspecial 2-groups and to braiding representations of the Ivanov type.
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Papers by Louis H Kauffman