Józef Henryk Przytycki
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Topological Aspects of Biosemiotics
tripleC 5(2): 49-63, 2007 ISSN 1726-670X http://tripleC.uti.at Topological Aspects of Biosemiotics Rainer E. Zimmermann IAG Philosophische Grundlagenprobleme, U Kassel / Clare Hall, UK – Cambridge / Lehrgebiet Philosophie, FB 13 AW, FH Muenchen, Lothstr. 34, D – 80335 München E-mail: [email protected] Abstract: According to recent work of Bounias and Bonaly cal terms with a view to the biosemiotic consequences. As this (2000), there is a close relationship between the conceptualiza- approach fits naturally into the Kassel programme of investigat- tion of biological life and mathematical conceptualization such ing the relationship between the cognitive perceiving of the that both of them co-depend on each other when discussing world and its communicative modeling (Zimmermann 2004a, preliminary conditions for properties of biosystems. More pre- 2005b), it is found that topology as formal nucleus of spatial cisely, such properties can be realized only, if the space of modeling is more than relevant for the understanding of repre- orbits of members of some topological space X by the set of senting and co-creating the world as it is cognitively perceived functions governing the interactions of these members is com- and communicated in its design. Also, its implications may well pact and complete. This result has important consequences for serve the theoretical (top-down) foundation of biosemiotics the maximization of complementarity in habitat occupation as itself. well as for the reciprocal contributions of sub(eco)systems with respect -
Ph.D. University of Iowa 1983, Area in Geometric Topology Especially Knot Theory
Ph.D. University of Iowa 1983, area in geometric topology especially knot theory. Faculty in the Department of Mathematics & Statistics, Saint Louis University. 1983 to 1987: Assistant Professor 1987 to 1994: Associate Professor 1994 to present: Professor 1. Evidence of teaching excellence Certificate for the nomination for best professor from Reiner Hall students, 2007. 2. Research papers 1. Non-algebraic killers of knot groups, Proceedings of the America Mathematical Society 95 (1985), 139-146. 2. Algebraic meridians of knot groups, Transactions of the American Mathematical Society 294 (1986), 733-747. 3. Isomorphisms and peripheral structure of knot groups, Mathematische Annalen 282 (1988), 343-348. 4. Seifert fibered surgery manifolds of composite knots, (with John Kalliongis) Proceedings of the American Mathematical Society 108 (1990), 1047-1053. 5. A note on incompressible surfaces in solid tori and in lens spaces, Proceedings of the International Conference on Knot Theory and Related Topics, Walter de Gruyter (1992), 213-229. 6. Incompressible surfaces in the knot manifolds of torus knots, Topology 33 (1994), 197- 201. 7. Topics in Classical Knot Theory, monograph written for talks given at the Institute of Mathematics, Academia Sinica, Taiwan, 1996. 8. Bracket and regular isotopy of singular link diagrams, preprint, 1998. 1 2 9. Regular isotopy of singular link diagrams, Proceedings of the American Mathematical Society 129 (2001), 2497-2502. 10. Normal holonomy and writhing number of polygonal knots, (with James Hebda), Pacific Journal of Mathematics 204, no. 1, 77 - 95, 2002. 11. Framing of knots satisfying differential relations, (with James Hebda), Transactions of the American Mathematics Society 356, no. -
Arxiv:Math/0307077V4
Table of Contents for the Handbook of Knot Theory William W. Menasco and Morwen B. Thistlethwaite, Editors (1) Colin Adams, Hyperbolic Knots (2) Joan S. Birman and Tara Brendle, Braids: A Survey (3) John Etnyre Legendrian and Transversal Knots (4) Greg Friedman, Knot Spinning (5) Jim Hoste, The Enumeration and Classification of Knots and Links (6) Louis Kauffman, Knot Diagramitics (7) Charles Livingston, A Survey of Classical Knot Concordance (8) Lee Rudolph, Knot Theory of Complex Plane Curves (9) Marty Scharlemann, Thin Position in the Theory of Classical Knots (10) Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory arXiv:math/0307077v4 [math.GT] 26 Nov 2004 A SURVEY OF CLASSICAL KNOT CONCORDANCE CHARLES LIVINGSTON In 1926 Artin [3] described the construction of certain knotted 2–spheres in R4. The intersection of each of these knots with the standard R3 ⊂ R4 is a nontrivial knot in R3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [86] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S3: knots K and J are equivalent if K# − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched. -
Various Topics in Rack and Quandle Homology
Various topics in rack and quandle homology April 14, 2010 Master's thesis in Mathematics Jorik Mandemaker supervisor: Dr. F.J.-B.J. Clauwens second reader: Prof. Dr. F. Keune student number: 0314145 Faculty of Science (FNWI) Radboud University Nijmegen Contents 1 Introduction 5 1.1 Racks and quandles . 6 1.2 Quandles and knots . 7 1.3 Augmented quandles and the adjoint group . 8 1.3.1 The adjoint group . 9 1.4 Connected components . 9 1.5 Rack and quandle homology . 10 1.6 Quandle coverings . 10 1.6.1 Extensions . 11 1.6.2 The fundamental group of a quandle . 12 1.7 Historical notes . 12 2 The second cohomology group of alexander quandles 14 2.1 The adjoint group of Alexander quandles . 15 2.1.1 Connected Alexander quandles . 15 2.1.2 Alexander quandles with connected components . 18 2.2 The second cohomology group of connected Alexander quandles . 19 3 Polynomial cocycles 22 3.1 Introduction . 23 3.2 Polynomial cochains . 23 3.3 A decomposition and filtration of the complex . 24 3.4 The 3-cocycles . 25 3.5 Variable Reduction . 27 3.6 Proof of surjectivity . 28 3.6.1 t = s .............................. 29 3.6.2 t < s .............................. 30 3.7 Final steps . 33 4 Rack and quandle modules 35 4.1 Rack and Quandle Modules . 36 4.1.1 Definitions . 36 4.1.2 Some examples . 38 4.2 Reduced rack modules . 39 1 4.3 Right modules and tensor products . 40 4.4 The rack algebra . 42 4.5 Examples . 44 4.5.1 A free resolution for trivial quandles . -
Notices of the American Mathematical Society June/July 2006
of the American Mathematical Society ... (I) , Ate._~ f.!.o~~Gffi·u. .4-e.e..~ ~~~- •i :/?I:(; $~/9/3, Honoring J ~ rt)d ~cLra-4/,:e~ o-n. /'~7 ~ ~<A at a Gift from fL ~ /i: $~ "'7/<J/3. .} -<.<>-a.-<> ~e.Lz?-1~ CL n.y.L;; ro'T>< 0 -<>-<~:4z_ I Kumbakonam li .d. ~ ~~d a. v#a.d--??">ovt<.·c.-6 ~~/f. t:JU- Lo,.,do-,......) ~a page 640 ~!! ?7?.-L ..(; ~7 Ca.-uM /3~~-d~ .Y~~:Li: ~·e.-l a:.--nd '?1.-d- p ~ .di.,r--·c/~ C(c£~r~~u . J~~~aq_ f< -e-.-.ol ~ ~ ~/IX~ ~ /~~ 4)r!'a.. /:~~c~ •.7~ The Millennium Grand Challenge .(/.) a..Lu.O<"'? ...0..0~ e--ne_.o.AA/T..C<.r~- /;;; '7?'E.G .£.rA-CLL~ ~ ·d ~ in Mathematics C>n.A..U-a.A-d ~~. J /"-L .h. ?n.~ ~?(!.,£ ~ ~ &..ct~ /U~ page 652 -~~r a-u..~~r/a.......<>l/.k> 0?-t- ~at o ~~ &~ -~·e.JL d ~~ o(!'/UJD/ J;I'J~~Lcr~~ 0 ??u£~ ifJ>JC.Qol J ~ ~ ~ -0-H·d~-<.() d Ld.orn.J,k, -F-'1-. ~- a-o a.rd· J-c~.<-r:~ rn-u-{-r·~ ~'rrx ~~/ ~-?naae ~~ a...-'XS.otA----o-n.<l C</.J.d:i. ~~~ ~cL.va- 7 ??.L<A) ~ - Ja/d ~~ ./1---J- d-.. ~if~ ~0:- ~oj'~ t1fd~u: - l + ~ _,. :~ _,. .~., -~- .. =- ~ ~ d.u. 7 ~'d . H J&."dIJ';;;::. cL. r ~·.d a..L- 0.-n(U. jz-o-cn-...l- o~- 4; ~ .«:... ~....£.~.:: a/.l~!T cLc.·£o.-4- ~ d.v. /-)-c~ a;- ~'>'T/JH'..,...~ ~ d~~ ~u ~ ~ a..t-4. l& foLk~ '{j ~~- e4 -7'~ -£T JZ~~c~ d.,_ .&~ o-n ~ -d YjtA:o ·C.LU~ ~or /)-<..,.,r &-. -
The Homotopy Types of Free Racks and Quandles
The homotopy types of free racks and quandles Tyler Lawson and Markus Szymik June 2021 Abstract. We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor’s theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, more complicated than what any Wirtinger presentation coming from a diagram predicts. 1 Introduction Racks and quandles form two algebraic theories that are closely related to groups and symmetry. A rack R has a binary operation B such that the left-multiplications s 7! r Bs are automorphism of R for all elements r in R. This means that racks bring their own symmetries. All natural sym- metries, however, are generated by the canonical automorphism r 7! r B r (see [45, Thm. 5.4]). A quandle is a rack for which the canonical automorphism is the identity. Every group defines −1 a quandle via conjugation g B h = ghg , and so does every subset closed under conjugation. The most prominent applications of these algebraic concepts so far are to the classification of knots, first phrased in terms of quandles by Joyce [28, Cor. 16.3] and Matveev [35, Thm. -
Prospects in Topology
Annals of Mathematics Studies Number 138 Prospects in Topology PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER edited by Frank Quinn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1995 Copyright © 1995 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 987654321 Library of Congress Cataloging-in-Publication Data Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751 The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY F r a n k Q u in n , E d it o r Proceedings of a conference in honor of William Browder Princeton, March 1994 Contents Foreword..........................................................................................................vii Program of the conference ................................................................................ix Mathematical descendants of William Browder...............................................xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay........................................................................3 A. -
Quandles an Introduction to the Algebra of Knots
STUDENT MATHEMATICAL LIBRARY Volume 74 Quandles An Introduction to the Algebra of Knots Mohamed Elhamdadi Sam Nelson Quandles An Introduction to the Algebra of Knots http://dx.doi.org/10.1090/stml/074 STUDENT MATHEMATICAL LIBRARY Volume 74 Quandles An Introduction to the Algebra of Knots Mohamed Elhamdadi Sam Nelson American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 57M25, 55M25, 20N05, 20B05, 55N35, 57M05, 57M27, 20N02, 57Q45. For additional information and updates on this book, visit www.ams.org/bookpages/stml-74 Library of Congress Cataloging-in-Publication Data Elhamdadi, Mohamed, 1968– Quandles: an introduction to the algebra of knots / Mohamed Elhamdadi, Sam Nelson. pages cm. – (Student mathematical library ; volume 74) Includes bibliographical references and index. ISBN 978-1-4704-2213-4 (alk. paper) 1. Knot theory. 2. Low-dimensional topology. I. Nelson, Sam, 1974– II. Title. III. Title: Algebra of Knots. QA612.2.E44 2015 514.2242–dc23 2015012551 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. -
INTRODUCTION to ALGEBRAIC THEORY of QUANDLES Lecture 1
INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLES Lecture 1 Valeriy Bardakov Sobolev Institute of Mathematics, Novosibirsk August 24-28, 2020 ICTS Bengaluru, India V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 1 / 53 Green-board in ICTS V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 2 / 53 Quandles A quandle is a non-empty set with one binary operation that satisfies three axioms. These axioms motivated by the three Reidemeister moves of diagrams 3 of knots in the Euclidean space R . Quandles were introduced independently by S. Matveev and D. Joyce in 1982. V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 3 / 53 Definition of rack and quandle Definition A rack is a non-empty set X with a binary algebraic operation (a; b) 7! a ∗ b satisfying the following conditions: (R1) For any a; b 2 X there is a unique c 2 X such that a = c ∗ b; (R2) Self-distributivity: (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a; b; c 2 X. A quandle X is a rack which satisfies the following condition: (Q1) a ∗ a = a for all a 2 X. V. Bardakov (Sobolev Institute of Math.) INTRODUCTION TO ALGEBRAIC THEORY OF QUANDLESAugust 27-30, 2019 4 / 53 Symmetries Let X be a rack. For any a 2 X define a map Sa : X −! X, which sends any element x 2 X to element x ∗ a, i. e. xSa = x ∗ a. We shall call Sa the symmetry at a. -
Scottish Physics and Knot Theory's Odd Origins
Scottish Physics and Knot Theory’s Odd Origins Daniel S. Silver My mother, a social worker and teacher, encouraged my interest in the mysteries of thought. My father, a physical chemist, fostered my appreciation of the history of science. The former chair of my department, prone to unguarded comment, once accused me of being too young to think about such things. As the years pass, I realize how wrong he is. Introduction. Knot theory is one of the most active research areas of mathematics today. Thousands of refereed articles about knots have been published during just the past ten years. One publication, Journal of Knot Theory and its Ramifications, published by World Scientific since 1992, is devoted to new results in the subject. It is perhaps an oversimplification, but not a grave one, to say that two mildly eccentric nineteenth century Scottish physicists, William Thomson (a.k.a. Lord Kelvin) and Peter Guthrie Tait were responsible for modern knot theory. Our purpose is not to debate the point. Rather it is to try to understand the beliefs and bold expectations that motivated Thomson and Tait. James Clerk Maxwell’s influence on his friend Tait is well known. However, Maxwell’s keen interest in the developing theory of knots and links has become clear only in recent years, thanks in great part to the efforts of M. Epple. Often Maxwell was a source of information and humorous encouragement. At times he attempted to restrain Tait’s flights from the path of science. Any discussion of the origin of knot theory requires consideration of the colorful trefoil of Scottish physicists: Thomsom, Tait and Maxwell. -
The Catalan Mathematical Society EMS June 2000 3 EDITORIAL
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 36 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] KRZYSZTOF CIESIELSKI June 2000 Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland EMS News : Agenda, Editorial, 3ecm, Bedlewo Meeting, Limes Project ........... 2 e-mail: [email protected] KATHLEEN QUINN Open University [address as above] Catalan Mathematical Society ........................................................................... 3 e-mail: [email protected] SPECIALIST EDITORS The Hilbert Problems ....................................................................................... 10 INTERVIEWS Steen Markvorsen [address as above] SOCIETIES Interview with Peter Deuflhard ....................................................................... 14 Krzysztof Ciesielski [address as above] EDUCATION Vinicio Villani Interview with Jaroslav Kurzweil ..................................................................... 16 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy A Major Challenge for Mathematicians ........................................................... 20 e-mail: [email protected] MATHEMATICAL PROBLEMS Paul Jainta EMS Position Paper: Towards a European Research Area ............................. 24 -
The Catalan Mathematical Society EMS June 2000 3 EDITORIAL
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS STEEN MARKVORSEN Department of Mathematics Technical University of Denmark Building 303 NEWSLETTER No. 36 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] KRZYSZTOF CIESIELSKI June 2000 Mathematics Institute Jagiellonian University Reymonta 4 30-059 Kraków, Poland EMS News : Agenda, Editorial, 3ecm, Bedlewo Meeting, Limes Project ........... 2 e-mail: [email protected] KATHLEEN QUINN Open University [address as above] Catalan Mathematical Society ........................................................................... 3 e-mail: [email protected] SPECIALIST EDITORS The Hilbert Problems ....................................................................................... 10 INTERVIEWS Steen Markvorsen [address as above] SOCIETIES Interview with Peter Deuflhard ....................................................................... 14 Krzysztof Ciesielski [address as above] EDUCATION Vinicio Villani Interview with Jaroslav Kurzweil ..................................................................... 16 Dipartimento di Matematica Via Bounarotti, 2 56127 Pisa, Italy A Major Challenge for Mathematicians ........................................................... 20 e-mail: [email protected] MATHEMATICAL PROBLEMS Paul Jainta EMS Position Paper: Towards a European Research Area ............................. 24