arXiv subjects

Louis H. Kauffman

Publications and source records attributed to Louis H. Kauffman.

At least 73 records · Page 4Linked to original sources

arXiv2018

Topological surgery in cosmic phenomena

We connect topological changes that can occur in $3$-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the $3$-dimensional spatial section of spacetime, we describe the changes of topology occurring in these processes by determining the resulting $3$-manifold and its fundamental group. As these global changes are induced by local processes, we use the local form of Morse functions to provide an algebraic formulation of their temporal evolution and propose a potential energy function which, in some cases, could give rise to the local forces related to surgery. We further show how this topological perspective gives new insight for natural phenomena exhibiting surgery, in all dimensions, while emphasizing the $3$-dimensional case, which describes cosmic phenomena. This work makes new bridges between topology and natural sciences and creates a platform for exploring geometrical physics.

math.GT↗

arXiv2018

Detecting and visualizing 3-dimensional surgery

Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this expository paper, we present two different approaches for understanding and visualizing the process of $3$-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how $3$-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations for both types of $3$-dimensional surgery as different rotations of the decompactified $2$-sphere. Each rotation produces a different decomposition of the $3$-sphere which corresponds to a different visualization of the $4$-dimensional process of $3$-dimensional surgery.

math.GT↗

arXiv2018

Black holes and topological surgery

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

math.GT↗

arXiv2018

The generalized Yamada polynomials of virtual spatial graphs

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is similar to the topological definition of a virtual link. Our main goal is to generalize the classical Yamada polynomial that is defined for a spatial graph. We define a generalized Yamada polynomial for a virtual spatial graph and prove that it can be normalized to a rigid vertex isotopic invariant and to a pliable vertex isotopic invariant for graphs with maximum degree at most 3. We consider the connection and difference between the generalized Yamada polynomial and the Dubrovnik polynomial of a classical link. The generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for virtual links such that it can be used to detect the non-classicality of some virtual links. We obtain a specialization for the generalized Yamada polynomial (via the Jones-Wenzl projector $P_2$ acting on a virtual spatial graph diagram), that can be used to write a program for calculating it based on Mathematica Code.

math.GT↗

arXiv2018

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends only on $K$. Furthermore we prove the following: The submanifolds, $Q$ and the embedded torus made from $K,$ defined by Satoh's method, in $S^4$ are isotopic. We succeed to generalize the above construction to the virtual 2-knot case. Note that Satoh's method says nothing about the virtual 2-knot case. Rourke's interpretation of Satoh's method is that one puts `fiber-circles' on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circles to each point of the virtual 2-knot diagram. We prove the following: If a virtual 2-knot diagram $α$ has a virtual branch point, $α$ cannot be covered by such fiber-circles. We obtain a new equivalence relation, the $\mathcal E$-equivalence relation of the set of virtual 2-knot diagrams, by using our spinning construction. We prove that there are virtual 2-knot diagrams that are virtually nonequivalent but are $\mathcal E$-equivalent. Although Rourke claimed that two virtual 1-knot diagrams $α$ and $β$ are fiberwise equivalent if and only if $α$ and $β$ are welded equivalent, we state that this claim is wrong. We prove that two virtual 1-knot diagrams $α$ and $β$ are fiberwise equivalent if and only if $α$ and $β$ are rotational welded equivalent.

math.GT↗

arXiv2018

Topological Aspects of Quantum Entanglement

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condition for forming invariants of knots from braid closures via solutions to the Yang-Baxter Equation. We show that the arguments used by these authors generalize to essentially the same results for quantum invariant state summation models of knots. We also given an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing.The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement, and the ER = EPR hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, 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.

math.GT↗

arXiv2018

Edge colourings and topological graph polynomials

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.

math.CO↗

arXiv2018

Non-Commutative Worlds and Classical Constraints

This paper reviews results about discrete physics and non-commutative worlds and explores further the structure and consequences of constraints linking classical calculus and discrete calculus formulated via commutators. In particular we review how the formalism of generalized non-commutative electromagnetism follows from a first order constraint and how, via the Kilmister equation, relationships with general relativity follow from a second order constraint. It is remarkable that a second order constraint, based on interlacing the commutative and non-commutative worlds, leads to an equivalent tensor equation at the pole of geodesic coordinates for general relativity.

math-ph↗

arXiv2018

Brieskorn submanifolds, Local moves on knots, and knot products

We prove the following: Let $2p + 1$ be no less than 5 and $p$ be a natural number. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional connected, $(p-1)$-connected, simple submanifolds of the standard $(2p+3)$-sphere. Then $K$ is equivalent to $J$ if and only if a Seifert matrix associated with a simple Seifert hypersurface for $K$ is $(-1)^p$-$S$-equivalent to that for $J$. We also discuss the $2p+1=3$ case. This result implies one of our main results: Let $μ$ be a natural number. A 1-link $A$ is pass-move equivalent to a 1-link $B$ if and only if the knot product of $A$ and $μ$ copies of the Hopf link is $(2μ+1, 2μ+1)$-pass-move equivalent to that of B and $μ$ copies of the Hopf link. It also implies the other of them: Two-fold cyclic suspension commutes with the performance of the twist move for spherical $(2k+1)$-knots ($2k+1>4$). Furthemroe we prove the following: Let $2p+1$ be no less than 5 and p be a natural number. Let $K$ be a closed oriented $(2p+1)$-dimensionalsubmanifold of the standard $(2p+3)$-sphere. Then $K$ is a Brieskorn submanifold if and only if $K$ is connected, $(p-1)$-connected, simple and has a $(p+1)$-Seifert matrix associated with a simple Seifert hypersurface that is $(-1)^p$-$S$-equivalent to a Kauffman-Neumann-type, or a KN-type (See the body of the paper for a definition.) We also discuss the $2p+1=3$ case.

math.GT↗

arXiv2018

Colorings beyond Fox: the other linear Alexander quandles

This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then so does any other diagram of the same knot. This is called p-colorability. It is also associated to systems of linear homogeneous equations over the residues mod p, by regarding the arcs of the diagram as variables and assigning the equation "twice the over-arc minus the sum of the under-arcs equals zero" to each crossing. The knot invariant is here the existence or non-existence of non-trivial solutions of these systems of equations, when working over the integers modulo p (a non-trivial solution is such that not all variables take up the same value). Another knot invariant is the minimum number of distinct colors (values) these non-trivial solutions require, should they exist. This corresponds to finding a basis, supported by a diagram, in which these solutions take up the least number of distinct values. The actual minimum is hard to calculate, in general. For the first few primes, less than 17, it depends only on the prime, p, and not on the specific knots that admit non-trivial solutions, modulo p. For primes larger than 13 this is an open problem. In this article, we begin the exploration of other generalizations of these colorings (which also involve systems of linear homogeneous equations mod p) and we give lower bounds for the number of colors.

math.GT↗

arXiv2018

Local moves on knots and products of knots II

We use the terms, knot product and local move, as defined in the text of the paper. Let $n$ be an integer$\geqq3$. Let $\mathcal S_n$ be the set of simple spherical $n$-knots in $S^{n+2}$. Let $m$ be an integer$\geqq4$. We prove that the map $j:\mathcal S_{2m}\to\mathcal S_{2m+4}$ is bijective, where $j(K)=K\otimes$Hopf, and Hopf denotes the Hopf link. Let $J$ and $K$ be 1-links in $S^3$. Suppose that $J$ is obtained from $K$ by a single pass-move, which is a local-move on 1-links. Let $k$ be a positive integer. Let $P\otimes^kQ$ denote the knot product $P\otimes\underbrace{Q\otimes...\otimes Q}_k$. We prove the following: The $(4k+1)$-dimensional submanifold $J\otimes^k{\rm Hopf}$ $\subset S^{4k+3}$ is obtained from $K\otimes^k{\rm Hopf}$ by a single $(2k+1,2k+1)$-pass-move, which is a local-move on $(4k+1)$-submanifolds contained in $S^{4k+3}$. See the body of the paper for the definitions of all local moves in this abstract. We prove the following: Let $a,b,a',b'$ and $k$ be positive integers. If the $(a,b)$ torus link is pass-move equivalent to the $(a',b')$ torus link, then the Brieskorn manifold $Σ(a,b,\underbrace{2,...,2}_{2k})$ is diffeomorphic to $Σ(a',b',\underbrace{2,...,2}_{2k})$ as abstract manifolds. Let $J$ and $K$ be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in $S^4$. Suppose that $J$ is obtained from $K$ by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in $S^4$. Let $k$ be an integer$\geq2$. We prove the following: The $(4k+2)$-submanifold $J\otimes^k{\rm Hopf}$ $\subset S^{4k+4}$ is obtained from $K\otimes^k{\rm Hopf}$ by a single $(2k+1,2k+2)$-pass-move, which is a local-move on $(4k+2)$-dimensional submanifolds contained in $S^{4k+4}$.

math.GT↗

arXiv2017

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Huygens triviality-a concept invented by Jacques Hadamard-describes an equivalence class connecting those 2nd order partial differential equations which are transformable into the wave equation. In this work it is demonstrated, that the Schrödinger equation with the time-independent Hamiltonian belongs to such an equivalence class. The wave equation is the equation for which Huygens' principle (HP) holds. The HP was a subject of confusion in both physics and mathematics literature for a long time. Not surprisingly, the role of this principle was obscured from the beginnings of quantum mechanics causing some theoretical and experimental misunderstandings. The purpose of this work is to bring the full clarity into this topic. By doing so, we obtained a large amount of new results related to uses of Lie sphere geometry, of twistors, of Dupin cyclides, of null electromagnetic fields, of AdS-CFT correspondence, of Penrose limits, of geometric algebra, etc. in physical problems ranging from the atomic to high energy physics and cosmology.

physics.gen-ph↗

arXiv2017

Infinitely many prime knots with the same Alexander invariants

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels.

math.GT↗

arXiv2017

New Invariants of Knotoids

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding invariants of virtual knots. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a classical knotoid. We show that classical knotoids have symmetric affine index polynomials.

math.GT↗

arXiv2017

Investigation of bifurcations in the behavior of the process equation

This paper investigates the different behaviors of the process equation and parameters of their occurrences. The process equation is a multistable one dimensional map with nonlinear feedback and can show various behaviors such as period doubling route to chaos, bios, unstable windows and periodic windows. In this note, we focus on different behaviors of the process equation by a deep look at phase portraits and cobweb plots. Therefore, period doubling route to chaos and unifurcations of the equation are investigated, and also the parameter of its entrance to biotic pattern is discussed. The control parameter for the process equation is g (the coupling constant). In higher g, the system shows some unstable and periodic windows among the biotic behaviors. Different patterns of these windows and the reasons of their happenings are investigated mathematically. In addition, some other types of unstable and periodic windows are discussed and Q-curves determine the difference of these windows with the previous ones.

math.DS↗

arXiv2016

A State Calculus for Graph Coloring

This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs presented as immersions in the plane.

math.CO↗

arXiv2016

Recursive Distinctioning

Recursive distinctioning (RD) is a name coined by Joel Isaacson in his original patent document describing how fundamental patterns of process arise from the systematic application of operations of distinction and description upon themselves. Recursive distinctioning means just what it says. A pattern of distinctions is given in a space based on a graphical structure (such as a line of print or a planar lattice or given graph). Each node of the graph is occupied by a letter from some arbitrary alphabet. A specialized alphabet is given that can indicate distinctions about neighbors of a given node. The neighbors of a node are all nodes that are connected to the given node by edges in the graph. The letters in the specialized alphabet (call it SA) are used to describe the states of the letters in the given graph and at each stage in the recursion, letters in SA are written at all nodes in the graph, describing its previous state. The recursive structure that results from the iteration of descriptions is called recursive distinctioning.

physics.gen-ph↗

arXiv2016

Braiding Majorana Fermions

In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structure of collectivies of electrons, as in the quantum Hall effect, but also in the structure of single electrons both by experiments with electrons in nanowires and also by the decomposition of the operator algebra for a fermion into a Clifford algebra generated by two Majorana operators. The purpose of this paper is to discuss these braiding representations, important for relationships among physics, quantum information and topology. A new result in this paper is the Clifford Braiding Theorem. This theorem shows that the Majorana operators give rise to a particularly robust representation of the braid group that is then further represented to find the phases of the fermions under their exchanges in a plane space. The more robust representation in our braiding theorem will be the subject of further work.

cond-mat.str-el↗