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Louis H Kauffman was born February 3, 1945 in Potsdam, New York. He graduated in 1962 from Norwood, Norfolk High Scho... moreLouis H Kauffman was born February 3, 1945 in Potsdam, New York. He graduated in 1962 from Norwood, Norfolk High School and earned a BS in Mathematics from MIT in 1966. He received a PhD in Mathematics from Princeton University in 1972. He teaches at the University of Illinois from 1971 to the present time, where is now Professor of Mathematics Emeritus. Kauffman is known for introducing state summations into the theory of invariants of knots, beginning with his model for the Alexander Polynomial and his bracket state summation model for the Jones polynomial. He is know for his discovery of the two-variable Kauffman polynomial and for his discovery and investigation of virtual knot theory. He is the author of four books on the theory of knots and the editor of a number of collections of papers related to knot theory. Kauffman is the Editor in Chief and founding editor of the Journal of Knot Theory and Its Ramifications (JKTR) published by World Scientific. Kauffman is the editor of the Book Series on Knots and Everything published by World Scientific. Kauffman is a Fellow of the American Mathematical Society and a former Polya Lecturer for the Mathematical Society of America. Kauffman is the recipient of the Warren McCulloch Award and the Norbert Wiener Gold Medal of the American Society for Cybernetics. His research interests are in knot theory, quantum theory and the epistemology of form. He writes a regular column on Virtual Logic for the Journal, Cybernetics and Human Knowing. He plays clarinet in the Chicago based ChickenFat Klezmer Orchestra. edit
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Kauffman's PhD advisor is William Browderedit
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.... 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...
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... 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... 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 ...
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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... 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
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Mathematics and Isotopy
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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 how to construct a new topological space, that welds the network and the background space together. This construction embodies the principle... 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.
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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... 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.
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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... 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 other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend... 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 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 the end, we arrive at a summary formalism, a chapter in "boundary mathematics" where there are not only containers <> but... 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 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... 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 following properties: (a) is Turing complete, (b) has a model of decentralized, distributed computing associated to it, (c) works at the level of... 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
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:... 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...
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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... 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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This volume is a very well-formulated mathematical approach to systems theory based in set theory and category theory.
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... Klima, G. (nd). Website, Professor of Philosophy. Fordham University. Retrieved August 10, 2008 from http:// www.fordham.edu/gsas/phil/klima/ New American Standard Bible. (1995). Retrieved August 2008 from... more
... Klima, G. (nd). Website, Professor of Philosophy. Fordham University. Retrieved August 10, 2008 from http:// www.fordham.edu/gsas/phil/klima/ New American Standard Bible. (1995). Retrieved August 2008 from http://nasb.scripturetext.com/john/1.htm Russell, B. (1900). ...
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We introduce new topological quantum invariants of compact oriented 3manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form F×I where I... more
We introduce new topological quantum invariants of compact oriented 3manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form F×I where I denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. Since virtual knots and links are represented by embeddings of circles in thickened surfaces, we refer to embeddings of circles in the 3-sphere as classical links. Classical links are the same as virtual links that can be represented in a thickened 2-sphere and it is a fact that classical links, up to isotopy, embed in the collection of virtual links taken up to isotopy. We give a new invariant of clas...
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This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the... more
This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the presence of local unitary transformations. We show that this same R generates a new non-trivial invariant of braids, knots, and links. Other solutions of the Yang-Baxter Equation are also shown to be universal for quantum computation. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, teleportation, and the structure of braiding in a topological quantum field theory.
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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 verify a Khovanov homology theory for unoriented virtual links.... 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 verify 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 new unoriented Jones polynomial for virtual links, which is an invariant in the category of 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 definition for integral Khovanov homology is described. Finally, we define unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology. The K...
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The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the... more
The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten-Reshetikhin-Turaev invariant and trivial 3-manifold fundamental group.
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Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot... more
Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3-space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor's quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more "physics friendly" because, unlike the Reidemeister moves, they respect the differential geometry of 3-space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state s...
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In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of... more
In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L^2 paths x:[0,1]-->R^n in real n-space R^n. Also included is a curious discrete QHS algorithm which is dual to Shor's algorithm.
This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as... more
This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The entanglement criterion is used to explore this connection. The paper discusses non-locality in the light of this criterion.
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The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of... more
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wanderi...
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We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For... more
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.
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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.... more
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...
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This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots... more
This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangle fraction from the point of view of the bracket polynomial, accounts of the classification theorems for rational tangles and rational knots (and of our new proofs of these results) and a description of the use of rational tangles in studying DNA recombination.
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We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a... more
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T) on sublinks of a given link L, obtained by partitioning L into collections of sublinks.
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We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a... more
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T) on sublinks of a given link L, obtained by partitioning L into collections of sublinks.
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In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the... more
In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted K (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associated diagrams. For virtual knots, the new invariant, J(K), is a restriction of the writhe to the odd crossings of the diagram (A crossing is odd if it links an odd number of crossings in the Gauss code of the knot. The set of odd crossings is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies that K is non-trivial, non-classical and inequivalent to its planar mirror image. T...
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This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison it is fundamental to view topological entanglements such as braids as "entanglement... more
This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison it is fundamental to view topological entanglements such as braids as "entanglement operators" and to associate to them unitary operators that are capable of creating quantum entanglement.
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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... 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.
Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a... more
Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result and a generalised bracket polynomial, we develop methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial). As examples we show that, except for special cases, link diagrams with a single virtualization and link diagrams with a single virtual crossing are non-classical.
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This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and examples are discussed. Then the paper details extensions and also limitations... more
This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and examples are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a separate, network model for quantum evolution and measurement, where the background space is replaced by an evolving network. In this model there is an analog of the Aravind Hypothesis that promises to directly illuminate relationships between physics, topology and quantum knots.
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This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we... more
This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is an unknot. The key theorem shows that the numerator of the sum of two rational tangles [P/Q] and [R/S] is unknotted if and only if PS + QR has absolute value equal to 1. The paper uses these results in studying processive DNA recombination, finding minimal size unknot diagrams, generalizing to collapses to knots as well as to unknots, and in finding unknots with arbirarily high complexity. The paper is self-contained, with a review of the theory of rational tangles and a last section on relationships of the theme of the paper with other aspects of topology and number theory. The present version of the paper contains a correction to Figure 22 of the previous (and publi...
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We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find... more
We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.
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This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant... more
This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.
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We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful... more
We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for links in a 3-manifold other than the 3-sphere.
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of... more
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author and Heather Dye and Aaron Kaestner. Problems related to band-passing are explained, and a theory of isotopy of virtual surfaces is formulated in terms of a generalization of the Yoshikawa moves.
This is a handwritten introduction to the Temperley Lieb Category. The purpose of this paper is to demonstrate that by not using handwriting we are losing a large dimension of mathematical and communicative possibility.