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Louis Kauffman

University of Illinois at Chicago, Mathematics, Faculty Member

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Louis H Kauffman is Professor Emeritus at the University of Illinois at Chicago. His research is in topology, knot theory, topological quantum field theory, quantum computing and foundational ideas in mathematics and science including algebra, logic, form and cybernetics. He is known for the discovery of state summation models in knot theory and for the knot polynomial called the Kauffman polynomial. He is the originator of a significant generalization of classical knot theory called virtual knot theory. Kauffman is a Fellow of the American Mathematical Society and author of numerous books related to knot theory. He is the Editor in Chief and founding editor of the Journal of Knot Theory and Its Ramifications published by World Scientific. He is the editor of the Book Series in Knots and Everything published by World Scientific.Kauffman is a former Polya Lecturer for the Mathematical Association of America, past president of the American Society for Cybernetics and the recipient of the Warren McCulloch (1993) and Norbert Wiener Medal (2014) awards of the American Society for Cybernetics and the recipient of the Ludwig Bertalanffy Award in Complexity Thinking (2016). Kauffman plays clarinet in the ChickenFat Klezmer Orchestra in Chicago, Illinois, USA.
Address: Chicago, Illinois, United States

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Papers by Louis Kauffman

An Affine Index Polynomial Invariant of Virtual Knots

arXiv (Cornell University), Nov 7, 2012

This paper describes a polynomial invariant of virtual knots that is defined in terms of an integ... more This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,including its relation to previous invariants of this type and we show how to construct Vassiliev invariants from the same data. 2000 Mathematics Subject Classification. 57M27 .

The Links-Gould Invariant of Links

arXiv (Cornell University), Nov 23, 1998

We introduce and study in detail an invariant of (1, 1) tangles. This invariant, derived from a f... 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.

Statistical mechanics and the Jones polynomial

Contemporary mathematics, 1988

This paper is dedicated to Anatoly Libgober on his 60-th birthday.

Topological Aspects of Quantum Entanglement

arXiv (Cornell University), Nov 23, 2016

and explored the idea of understanding quantum entanglement (the non-local correlation of certain... more and 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 , it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang-Baxter Equation. We show that the arguments used by generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang-Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of . We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give 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

A Linking Number Definition of the Affine Index Polynomial and Applications

arXiv (Cornell University), Nov 7, 2012

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polyn... more 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.

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

arXiv (Cornell University), Aug 9, 2018

in S 4 made from classical 1-knots compose an important class of spherical 2-knots (respectively,... more in S 4 made from classical 1-knots compose an important class of spherical 2-knots (respectively, embedded tori) contained in S 4 . Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of a submanifold. The construction proceeds as follows: It has been known that there is a consistent way to make an embedded circle C contained in (a closed oriented surface F )×(a closed interval [0, 1]) from any virtual 1-knot K. Embed F in S 4 by an embedding map f . Let F also denote f (F ). We can regard the tubular neighborhood of Thus we obtain an embedded torus Q ⊂ S 4 . We prove the following: The embedding type Q in S 4 depends only on K, and does not depend on f . Furthermore, the submanifolds, Q and the embedded torus made from K, defined by Satoh's method, of S 4 are isotopic. We generalize this construction in the virtual 1-knot case, and we also succeed to make a consistent construction of one-dimensional-higher tubes from any virtual 2dimensional knot. 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. Furthermore 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 E-equivalence relation of the set of virtual 2-knot diagrams, that is much connected with the welded equivalence relation and our spinning construction. We prove that there are virtual 2-knot diagrams, J and K, that are virtually nonequivalent but are 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 (the definiton of rotational welded equivalence is given in the body of the paper).

The generalized Yamada polynomials of virtual spatial graphs

arXiv (Cornell University), Jun 17, 2018

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007... more 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 nonclassicality 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.

Graphical virtual links and a polynomial of signed cyclic graphs

arXiv (Cornell University), Dec 18, 2017

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial constr... more For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F [G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F [G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F [G].

Quantum computing and the Jones polynomial

Contemporary mathematics, 2002

This paper is an exploration of relationships between the Jones polynomial and quantum computing.... more This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open.

Free fermions and the Alexander-Conway polynomial

Communications in Mathematical Physics, Oct 1, 1991

We show how the Conway Alexander polynomial arises from the q deformation of (Z 2 graded) sl(n,n)... more We show how the Conway Alexander polynomial arises from the q deformation of (Z 2 graded) sl(n,n) algebras. In the simplest 5/(1,1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first show how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into the U q sl(ί, 1) R matrix by going to the exterior power algebra. Using a det = str identity, this allows us to recover the state model of [K2, 89] as well. We also show how the U q sl(l, 1) algebra describes free fermions "propagating" on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.

Permutation and Its Partial Transpose

International Journal of Quantum Information, Aug 1, 2007

Permutation and its partial transpose play important roles in quantum information theory. The Wer... more Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang-Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the "PPT" algebra which guides the construction of multipartite symmetric states. The virtual knot theory having permutation as a virtual crossing provides a topological language describing quantum computation having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley-Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang-Baxter equations; and virtual Temperley-Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the "ABPK" diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies nontrivial unitary braid representations with universal quantum gates, and derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

On Vassiliev Invariants of Virtual Knots

arXiv (Cornell University), Aug 22, 2022

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual k... more We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in . In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.

Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links

arXiv (Cornell University), Aug 30, 2021

We introduce new topological quantum invariants, surface link quantum invariants, of compact orie... more We introduce new topological quantum invariants, surface link quantum invariants, of compact oriented 3-manifolds 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 classical links in the 3-sphere in the following sense: Consider a link L in S 3 of two components. The complement of a tubular neighborhood of L is a manifold whose boundary consists in two copies of a torus. Our new invariants, surface link quantum invariants, apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained. Contents 9. Main theorem 27 10. Application to classical knot theory 29 11. Comments from a TQFT viewpoint 29 12. A Problem 30 References 30

Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links

Contemporary mathematics, 2003

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and... more

Khovanov-Lipshitz-Sarkar homotopy type for links in thickened surfaces and those in $S^3$ with new modulis

arXiv (Cornell University), Sep 19, 2021

We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov... more We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov homology of links in thickened surfaces indexed by moduli space systems. This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the content of the paper for the definition). The question whether different choices of moduli spaces lead to the same stable homotopy type is open.

On Filamentations and Virtual Knots

arXiv (Cornell University), Dec 19, 2001

Virtual Crossing Number and the Arrow Polynomial

arXiv (Cornell University), Oct 21, 2008

We introduce a new polynomial invariant of virtual knots and links and use this invariant to comp... more

Graphical Constructions for the sl(3), so(3) and G2 Invariants for Virtual Knots, Virtual Braids and Free Knots

arXiv (Cornell University), Jun 27, 2014

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. T... more We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and 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.

Virtual Knots and Links

arXiv (Cornell University), Feb 1, 2005

This paper is an introduction to the subject of virtual knot theory, combined with a discussion o... more 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 K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using Haken-Matveev technique, for recognizing virtual links. This paper may be read as both an introduction and as a research paper. For more about Haken-Matveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL].

Virtual Knot Theory

arXiv (Cornell University), Nov 5, 1998

This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Fr... more

An Affine Index Polynomial Invariant of Virtual Knots

arXiv (Cornell University), Nov 7, 2012

This paper describes a polynomial invariant of virtual knots that is defined in terms of an integ... more This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,including its relation to previous invariants of this type and we show how to construct Vassiliev invariants from the same data. 2000 Mathematics Subject Classification. 57M27 .

The Links-Gould Invariant of Links

arXiv (Cornell University), Nov 23, 1998

We introduce and study in detail an invariant of (1, 1) tangles. This invariant, derived from a f... 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.

Statistical mechanics and the Jones polynomial

Contemporary mathematics, 1988

This paper is dedicated to Anatoly Libgober on his 60-th birthday.

Topological Aspects of Quantum Entanglement

arXiv (Cornell University), Nov 23, 2016

and explored the idea of understanding quantum entanglement (the non-local correlation of certain... more and 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 , it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang-Baxter Equation. We show that the arguments used by generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang-Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of . We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give 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

A Linking Number Definition of the Affine Index Polynomial and Applications

arXiv (Cornell University), Nov 7, 2012

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polyn... more 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.

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

arXiv (Cornell University), Aug 9, 2018

in S 4 made from classical 1-knots compose an important class of spherical 2-knots (respectively,... more in S 4 made from classical 1-knots compose an important class of spherical 2-knots (respectively, embedded tori) contained in S 4 . Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of a submanifold. The construction proceeds as follows: It has been known that there is a consistent way to make an embedded circle C contained in (a closed oriented surface F )×(a closed interval [0, 1]) from any virtual 1-knot K. Embed F in S 4 by an embedding map f . Let F also denote f (F ). We can regard the tubular neighborhood of Thus we obtain an embedded torus Q ⊂ S 4 . We prove the following: The embedding type Q in S 4 depends only on K, and does not depend on f . Furthermore, the submanifolds, Q and the embedded torus made from K, defined by Satoh's method, of S 4 are isotopic. We generalize this construction in the virtual 1-knot case, and we also succeed to make a consistent construction of one-dimensional-higher tubes from any virtual 2dimensional knot. 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. Furthermore 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 E-equivalence relation of the set of virtual 2-knot diagrams, that is much connected with the welded equivalence relation and our spinning construction. We prove that there are virtual 2-knot diagrams, J and K, that are virtually nonequivalent but are 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 (the definiton of rotational welded equivalence is given in the body of the paper).

The generalized Yamada polynomials of virtual spatial graphs

arXiv (Cornell University), Jun 17, 2018

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007... more 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 nonclassicality 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.

Graphical virtual links and a polynomial of signed cyclic graphs

arXiv (Cornell University), Dec 18, 2017

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial constr... more For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F [G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F [G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F [G].

Quantum computing and the Jones polynomial

Contemporary mathematics, 2002

This paper is an exploration of relationships between the Jones polynomial and quantum computing.... more This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open.

Free fermions and the Alexander-Conway polynomial

Communications in Mathematical Physics, Oct 1, 1991

We show how the Conway Alexander polynomial arises from the q deformation of (Z 2 graded) sl(n,n)... more We show how the Conway Alexander polynomial arises from the q deformation of (Z 2 graded) sl(n,n) algebras. In the simplest 5/(1,1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first show how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into the U q sl(ί, 1) R matrix by going to the exterior power algebra. Using a det = str identity, this allows us to recover the state model of [K2, 89] as well. We also show how the U q sl(l, 1) algebra describes free fermions "propagating" on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.

Permutation and Its Partial Transpose

International Journal of Quantum Information, Aug 1, 2007

Permutation and its partial transpose play important roles in quantum information theory. The Wer... more Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang-Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the "PPT" algebra which guides the construction of multipartite symmetric states. The virtual knot theory having permutation as a virtual crossing provides a topological language describing quantum computation having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley-Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang-Baxter equations; and virtual Temperley-Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the "ABPK" diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies nontrivial unitary braid representations with universal quantum gates, and derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

On Vassiliev Invariants of Virtual Knots

arXiv (Cornell University), Aug 22, 2022

We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual k... more We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in . In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.

Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links

arXiv (Cornell University), Aug 30, 2021

We introduce new topological quantum invariants, surface link quantum invariants, of compact orie... more We introduce new topological quantum invariants, surface link quantum invariants, of compact oriented 3-manifolds 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 classical links in the 3-sphere in the following sense: Consider a link L in S 3 of two components. The complement of a tubular neighborhood of L is a manifold whose boundary consists in two copies of a torus. Our new invariants, surface link quantum invariants, apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained. Contents 9. Main theorem 27 10. Application to classical knot theory 29 11. Comments from a TQFT viewpoint 29 12. A Problem 30 References 30

Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links

Contemporary mathematics, 2003

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and... more

Khovanov-Lipshitz-Sarkar homotopy type for links in thickened surfaces and those in $S^3$ with new modulis

arXiv (Cornell University), Sep 19, 2021

We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov... more We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov homology of links in thickened surfaces indexed by moduli space systems. This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the content of the paper for the definition). The question whether different choices of moduli spaces lead to the same stable homotopy type is open.

On Filamentations and Virtual Knots

arXiv (Cornell University), Dec 19, 2001

Virtual Crossing Number and the Arrow Polynomial

arXiv (Cornell University), Oct 21, 2008

We introduce a new polynomial invariant of virtual knots and links and use this invariant to comp... more

Graphical Constructions for the sl(3), so(3) and G2 Invariants for Virtual Knots, Virtual Braids and Free Knots

arXiv (Cornell University), Jun 27, 2014

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. T... more We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and 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.

Virtual Knots and Links

arXiv (Cornell University), Feb 1, 2005

This paper is an introduction to the subject of virtual knot theory, combined with a discussion o... more 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 K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using Haken-Matveev technique, for recognizing virtual links. This paper may be read as both an introduction and as a research paper. For more about Haken-Matveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL].

Virtual Knot Theory

arXiv (Cornell University), Nov 5, 1998

This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Fr... more