Special Issue "Diagrams, Topology, Categories and Logic"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 April 2015)

Special Issue Editor

Guest Editor
Prof. Louis H. Kauffman

Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago, Chicago, IL 60607-7045, USA
Website | E-Mail
Phone: (312) 996-3066
Interests: geometric topology; classical knot theory; virtual knot theory; higher dimensional knot theory; quantum knots; topological quantum field theory; quantum computing; topological quantum computing; diagrammatic and categorical approaches to mathematical structure

Special Issue Information

Dear Colleagues,

The theme of this Special Issue is well-expressed by the following lines from the famous address made by David Hilbert to the International Congress of Mathematicians at Paris in 1900:

“To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Who could dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.”

In the intervening years since the turn of the 20th century, we have seen, particularly in logic, topology and category theory, the rise of many diagrammatic languages that are on the borderlines between different mathematical fields. Examples of such work are the Venn Diagrams of Boolean algebra, the diagrammatic logical calculi of Charles Sanders Peirce and George Spencer-Brown, the many reaches of the Theory of Graphs, the use of knot and link diagrams in knot theory and virtual knot theory, the Kirby Calculus using link diagrams to represent three and four dimensional manifolds, the wider range of drawings and languages of drawings that represent (formally) many problems in topology and differential geometry, the interrelationship of braided monoidal categories with many fields, the uses of circuit diagrams, categories, and quantum knot diagrams in quantum information theory, the new developments such as graphical lambda calculus  that relate topological and graphical languages to distributed computation, the use of tangle theory in low dimensional topology in studying the behavior of DNA replication, the use of formalisms in topology for wider mathematical, logical and computational issues.

This Special Issue invites papers on the use, creation and understanding of such diagrammatic geometric/topological/logical developments in mathematics and its applications to natural science and philosophy.

Prof. Louis H. Kauffman
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.



Keywords

  • concepts
  • signs
  • symbols
  • knot diagram
  • logic
  • topology
  • category theory
  • diagrammatic logical and topological calculi
  • graphical
  • lambda calculus
  • Feynman diagrams

Published Papers (5 papers)

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Research

Open AccessArticle Computing with Colored Tangles
Symmetry 2015, 7(3), 1289-1332; https://doi.org/10.3390/sym7031289
Received: 30 April 2015 / Revised: 6 July 2015 / Accepted: 14 July 2015 / Published: 20 July 2015
Cited by 3 | PDF Full-text (7621 KB) | HTML Full-text | XML Full-text
Abstract
We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated colored tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams represent bisimilar computations. We prove
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We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated colored tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams represent bisimilar computations. We prove that our model of computation is Turing complete and with bounded resources that it can decide any language in complexity class IP, sometimes with better performance parameters than corresponding classical protocols. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
Figures

Figure 1

Open AccessArticle Some Elementary Aspects of 4-Dimensional Geometry
Symmetry 2015, 7(2), 515-545; https://doi.org/10.3390/sym7020515
Received: 3 April 2015 / Revised: 27 April 2015 / Accepted: 27 April 2015 / Published: 4 May 2015
PDF Full-text (22965 KB) | HTML Full-text | XML Full-text
Abstract
We indicate that Heron’s formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in four-dimensional space. In the process of demonstrating this, we examine a number
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We indicate that Heron’s formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in four-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms and other elementary four-dimensional solids. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
Open AccessArticle Everywhere Equivalent 2-Component Links
Symmetry 2015, 7(2), 365-375; https://doi.org/10.3390/sym7020365
Received: 22 February 2015 / Revised: 26 March 2015 / Accepted: 30 March 2015 / Published: 13 April 2015
PDF Full-text (133 KB) | HTML Full-text | XML Full-text
Abstract
A link diagram is said to be (orientedly) everywhere equivalent if all the diagramsobtained by switching one crossing represent the same (oriented) link. We classify suchdiagrams of two components. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
Open AccessArticle Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis
Symmetry 2015, 7(2), 305-326; https://doi.org/10.3390/sym7020305
Received: 3 January 2015 / Revised: 14 March 2015 / Accepted: 19 March 2015 / Published: 31 March 2015
Cited by 1 | PDF Full-text (315 KB) | HTML Full-text | XML Full-text
Abstract
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck
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We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
Open AccessArticle Topology and the Visualization of Space
Symmetry 2015, 7(1), 32-39; https://doi.org/10.3390/sym7010032
Received: 20 October 2014 / Accepted: 19 December 2014 / Published: 30 December 2014
Cited by 3 | PDF Full-text (15369 KB) | HTML Full-text | XML Full-text
Abstract
Overlapping patterns provide the diagrammatics for four-dimensional space. If these patterns are three-dimensional lattices, and if one imagines them extended in three-dimensional space, then the diagram makes a model of physical space. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
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