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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

Invariants of Bonded Knotoids and Applications to Protein Folding

Symmetry

In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings... more In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological structure of proteins.

Determinants of rational knots

HAL (Le Centre pour la Communication Scientifique Directe), 2009

We study the Fox coloring invariants of rational knots. We express the propagation of the colors ... more

The prevalence of persistent tangles

Topology and its Applications, Feb 1, 2020

This article addresses persistent tangles. These are tangles whose presence in a knot diagram for... more This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article is that any knot admitting a non-trivial coloring gives rise to persistent tangles. Furthermore, we discuss when these persistent tangles are non-trivial.

Right integrals and invariants of three-manifolds

Geometry and Topology Monographs, Nov 18, 1999

This paper gives a summary of our approach to invariants of three manifolds via right integrals o... more

On the Maximum Number of Colors for Links

arXiv (Cornell University), May 7, 2012

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove tha... more 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.

Alexander-Conway Polynomial State Model and Link Homology

arXiv (Cornell University), Dec 10, 2014

This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is re... more

Colorings beyond Fox: The other linear Alexander quandles

Linear Algebra and its Applications, Jul 1, 2018

This article is about applications of linear algebra to knot theory. For example, for odd prime p... more 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.

Knot Logic: Logical Connection and Topological Connection

arXiv (Cornell University), Aug 25, 2015

The Delunification Process and Minimal Diagrams

arXiv (Cornell University), Jun 9, 2014

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

Vassiliev measures of complexity for open and closed curves in 3-space

arXiv (Cornell University), Apr 25, 2021

In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These a... more In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates and as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating selflinking integral. The double alternating self-linking integral is a topological invariant for closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities. For a polygonal curve with 4 edges, the double alternating self-linking integral coincides with the signed geometric probability of obtaining the knotoid k2.1 in a random projection direction.

Oriented Quantum Algebras and Invariants of Knots and Links

arXiv (Cornell University), Jun 3, 2000

oriented quantum algebras were motivated and introduced in a very natural categorical setting wit... more oriented quantum algebras were motivated and introduced in a very natural categorical setting within the context of knots and links, some examples were discussed and a rudimentary theory of oriented quantum algebras was sketched. Invariants of knots and links can be computed from oriented quantum algebras. Here we continue the study of oriented quantum algebras. We view them from a more algebraic perspective and develop a more detailed theory for them and their associated invariants. We study a class of examples associated with the HOMFLY polynomial in depth.

Upper Bounds for an Unknotting Sequence of Reidemeister Moves

arXiv (Cornell University), Jun 21, 2010

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknott... 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.

Rectangular knot diagrams classification with deep learning

arXiv (Cornell University), Nov 6, 2020

In this article we discuss applications of neural networks to recognising knots and, in particula... more In this article we discuss applications of neural networks to recognising knots and, in particular, to the unknotting problem. One of motivations for this study is to understand how neural networks work on the example of a problem for which rigorous mathematical algorithms for its solution are known. We represent knots by rectangular Dynnikov diagrams and apply neural networks to distinguish a given diagram's class from the given finite families of topological types. The data presented to the program is generated by applying Dynnikov moves to initial samples. The significance of using these diagrams and moves is that in this context the problem of determining whether a diagram is unknotted is a finite search of a bounded combinatorial space.

Vassiliev measures of complexity for open and closed curves in 3-space

In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These a... more In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates and as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant for closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities. For a polygonal curv...

Unknotting Unknots

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknott... 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.

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

Colorings beyond Fox : the other linear

This article is about applications of linear algebra to knot theory. For example, for odd prime p... more 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 t...

Vassiliev measures of complexity of open and closed curves in 3-space

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2021

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are... more In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

The prevalence of persistent tangles

Topology and its Applications, 2020

This article addresses persistent tangles. These are tangles whose presence in a knot diagram for... more This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article is that any knot admitting a non-trivial coloring gives rise to persistent tangles. Furthermore, we discuss when these persistent tangles are non-trivial.

Colorings beyond Fox: The other linear Alexander quandles

Linear Algebra and its Applications, 2018

This article is about applications of linear algebra to knot theory. For example, for odd prime p... more 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.