arXiv subjects

Louis Kauffman

Publications and source records attributed to Louis Kauffman.

12 recordsLinked to original sources

arXiv2025

Generalised Biracks and the Birack Polynomial Invariant

We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a class of knot theories amenable to a generalisation of theorem of Trace on regular homotopy. We take the opportunity to reprise the relevant generalised knot theory and the theory of generalised biracks in the light of this work and recent developments.

math.GT↗

arXiv2023

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian density inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

hep-th↗

arXiv2021

On the orbits associated with the Collatz conjecture

This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. Unfortunately, they could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.

math.CO↗

arXiv2019

Parity in Knotoids

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.

math.GT↗

arXiv2014

A Combinatorial Algorithm for Computing Higher Order Linking Numbers

We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the intersection theory in the process, which was first suggested by W. Massey. Massey introduced the higher-order linking as an application of his higher-order cohomology operations defined in terms of suitable cochains, and he calculated the third-order linking numbers for some 3-component links by shifting from cohomology and cup product to homology and intersection theory via duality theorems for manifolds. Later several works along this direction gave methods in various forms for computing the higher-order linking numbers, and others computed Milnor's invariants by using their well-known connection to Massey invariants; but a formal derivation of the general formulae in intersection theory which are used for the computation has not yet been given, and a concrete algorithm for the computation is hence lacking. In this paper we complete Massey's original approach by developing systematically the intersection theory at the relative chain-cochain level in the simplicial category, and use it to derive recursive combinatorial formulae for computing all higher-order linking numbers.

math.GT↗

arXiv2014

The Delunification Process and Minimal Diagrams

A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same number of colors. Thus any colored link diagram with a minimum number of colors (known as a minimal diagram) is equivalent to a colored lune-free diagram with that same number of colors. We call the passage from a link diagram to an equivalent lune-free diagram its delunification process. We then introduce a notion of grey sets in order to obtain higher lower bounds for minimum number of colors. We calculate these higher lower bounds for a number of prime moduli with the help of computer programs. For each number of crossings through 16, we list the lune-free diagrams and we color them. If the number of colors equals the corresponding higher lower bound we know we have a minimum number of colors. We also introduce and list the lune-free crossing number of a link i.e., the minimum number of crossings needed for a lune-free diagram of this link, and other related link invariants.

math.GT↗

arXiv2012

The Teneva Game

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

math.GT↗

arXiv2012

Emergent Braided Matter of Quantum Geometry

We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetravalent braided ribbon networks. Both schemes have been fruitful. The trivalent scheme has been quite successful at establishing a correspondence between braids and Standard Model particles, whereas the tetravalent scheme has naturally substantiated a rich, dynamical theory of interactions and propagation of braids, which is ruled by topological conservation laws. Some recent advances in the program indicate that the two schemes may converge to yield a fundamental theory of matter in quantum spacetime.

hep-th↗

arXiv2010

Oriented Quantum Algebras and Coalgebras, Invariants of Oriented 1-1 Tangles, Knots and Links

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras

math.QA↗

arXiv2009

NMR Quantum Calculations of the Jones Polynomial

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones Polynomial by nuclear-magnetic resonance (NMR), in addition we show how to escape from the limitations of NMR approaches that employ pseudo pure states. Specifically, we use two spin 1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the Trefoil Knot, the Figure Eight Knot and the Borromean Rings. After measuring the state of the molecule in each case, we are able to estimate the value of the Jones Polynomial for each of the knots.

quant-ph↗

arXiv2005

Virtual Biquandles

In the present paper, we describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle (Kauffman and Radford) the virtual quandle (Manturov), the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew, the concepts and properties of long virtual knots (Manturov), and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual knot invariants, give various presentations of it, and study several examples. Several conjectures and unsolved problems are presented throughout the paper.

math.GT↗

arXiv2005

Virtual Knots and Links

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a connected sum of two virtual knots $K_{1}$ and $K_{2}$ is trivial, then so are both $K_{1}$ and $K_{2}$. We establish an algorithm, using Haken-Matveev technique, for recognizing virtual knots. This paper may be read as both an introduction and as a research paper.

math.GT↗