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Louis H. Kauffman

Publications and source records attributed to Louis H. Kauffman.

At least 109 records · Page 6Linked to original sources
arXiv2014

Chemlambda, universality and self-multiplication

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 individual (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 languages.

cs.AI↗
arXiv2013

On the Maximum Number of Colors for Links

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove that there exists a non-trivial coloring of any Schubert Normal Form of it, modulo its determinant, which uses all colors available. If this determinant is an odd prime, then any non-trivial coloring of this Schubert Normal Form, modulo the determinant, uses all available colors. We prove also that the number of crossings in the Schubert Normal Form equals twice the determinant of the link minus 2. Facts about torus links and their coloring abilities are also proved.

math.GT↗
arXiv2013

Knot Logic and Topological Quantum Computing with Majorana Fermions

This paper is an introduction to relationships between quantum topology and quantum computing. We take a foundational approach, showing how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations and algebras of fermions. We show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion, a particle that is its own anti-particle and interacts with itself either to annihilate itself or to produce itself. We call negation in this mode the mark, as it operates on itself to change from marked to unmarked states. The mark viewed recursively as a simplest discrete dynamical system naturally generates the fermion algebra, the quaternions and the braid group representations related to Majorana fermions. The paper begins with these fundmentals. They provide a conceptual key to many of the models that appear later in the paper. In particular, the Fibonacci model for topological quantum computing is seen to be based on the fusion rules for a Majorana fermion and these in turn are the interaction rules for the mark seen as a logical particle. It requires a shift in viewpoint to see that the operator of negation can also be seen as a logical value. The quaternions emerge naturally from the reentering mark. All these models have their roots in unitary representations of the Artin braid group to the quaternions.

quant-ph↗
arXiv2012

A note on amphicheiral alternating links

After reconsidering the Dasbach-Hougardy counterexample to the Kauffman Conjecture on alternating knots, we reformulate the conjecture and consider Dasbach-Hougardy counterexample and similar counterexamples in the light of the reformulated conjecture.

math.GT↗
arXiv2012

A Linking Number Definition of the Affine Index Polynomial and Applications

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a Type 2 Vassiliev Invariant coming from the polynomial.

math.GT↗
arXiv2012

Nodal Parity Invariants of Knotted Rigid Vertex Graphs

This paper introduces new invariants of rigid vertex graph embeddings by using non-local combinatorial information that is available at each graphical node. The new non-local information that we use in this paper involves parity in the Gauss code of the underlying graph. We apply these methods to graphs in classical and virtual knot theory, and we give formulations for applications to the topology of protein folding.

math.AT↗
arXiv2012

Introduction to Virtual Knot Theory

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory both combinatorially and in terms of the knot theory in thickened surfaces. The arrow polynomial (of Dye and Kauffman) is a natural generalization of the Jones polynomial, obtained by using the oriented structure of diagrams in the state sum. The paper discusses uses of parity pioneered by Vassily Manturov and uses his parity bracket polynomial to give a counterexample to a conjecture of Fenn, Kauffman and Manturov. The paper gives an exposition of the categorification of the arrow polynomial due to Dye, Kauffman and Manturov and it gives one example (from many found by Aaron Kaestner) of a pair of virtual knots that are not distinguished by Khovanov homology (mod 2), or by the arrow polynomial, but are distinguished by a categorification of the arrow polynomial. Other examples of parity calculations are indicated. For example, this same pair is distinguished by the parity bracket polynomial.

math.GT↗
arXiv2012

A Categorical Model for the Virtual Braid Group

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang-Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang-Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

math.GT↗
arXiv2012

Following Knots Down Their Energy Gradients

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the trajectories of energy descent upon the initial geometric conditions of the knotted embedding.

math.GT↗
arXiv2011

Categorical Pairs and the Indicative Shift

This paper introduces the notion of a categorical pair, a pair of categories (C,C') such that every morphism in C is an object in C'. Categorical pairs are precursors to 2-categories. Arrows in C' can express relationships among the morphisms of C. In particular we show that by using a model of the linguistic process of naming, we can ensure that every morphism in C has an indirect self-reference of the form a -----> Fa where this arrow occurs in the category C'. This result is shown to generalize and clarify known fixed point theorems in logic and categories, and is applied to Goedel's Incompleteness Theorem, the Cantor Diagonal Process and the Lawvere Fixed Point Theorem. In particular we show that the indirect self-reference that is central to Goedel's Theorem is an instance of a general pattern here called the indicative shift.

math.CT↗
arXiv2011

Unknotting Unknots

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.

math.GT↗
arXiv2011

Parity, Skein Polynomials and Categorification

We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both polynomials. We present a number of examples found through our calculations. We provide tables of calculations for these invariants on virtual knots with at most 4 real crossings.

math.GT↗
arXiv2011

Eigenforms and Quantum Physics

This essay is a discussion of Heinz von Foerster's concept of eigenform and its relationship with the foundations of physics.

math-ph↗
arXiv2011

Quantizing Knots and Beyond

This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms for the Jones polynomial and relations with Khovanov homology. The purpose of this paper is to place such constructions in a general context of the quantization of mathematical structures.

quant-ph↗