Richard H Crowell Ralph H Fox r' Knot Theory ~ ~ ~ Sf l.O [jfl lArJ Springer-Verlag New York I kiddhcrg Berlin ~ & ~ I I R H Crowell R H Fox Department of l\:1athematics Dartmouth College Hanover, New Hampshire 03755 Formerly of Princeton University Princeton, New Jersey Editorial Board P R Halmos F W Gehring C C Moore ]'ianaging Editor Department of Mathematics University of California Santa Barbara, California 93106 Department of :Mathematics University of :Michigan Ann Arbor, J\:1ichigan 48104 Department of :Mathematics University of California at Berkeley Berkeley, California 94720 AMS Subject Classifications: 20E40, 55A05, 55A25, 55A30 Library of Congress Cat.aloging in Publication Data Crowell, Richard H Introduction to knot theory (Graduate texts in mathematics 57) Bibliography: p Includes index Knot theory I Fox, Ralph Hartzler, 1913joint author II Title III Series QA612.2.C76 1977 514'.224 77 -22776 ISBN All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Verlag © I B63 by R H Crowell and C Fox Pl'illtod ill tho United States of America IKBN :~K7 HO~7~·' Npl'illgOI'-Vn"'Hg Now VOI'I Chapter IV Presentation of Groups Introduction I Development of the presentation concept Presentations and presentation types :1 'rhe Tietze theorem Word subgroups and the associated homomorphiRm~ !) Free abel ian groups Calculation of Fundamental Groups I ntl'oduction I ItptntetionH and d('forrnat.ions ~ Ilornotop.v typo :1 'I'hp van K,unppft tht'on'rJl 13 14 The Free Groups Introduction The free group F[d'] Reduced words :1- Free groups Chapter V The Fundamental Group Introduction I Paths and loops Classes of paths and loops Change of basepoint Induced homomorphisms of fundamental groups :Fundamental group of the circle Chapter m CONTENTS X Chapter VI Presentation of a Knot Group Intl'oduction 'rhe over and under presentations 'rhe over and under presentations, continued 'rho Wirtinger presentation Examples of presentations Existence of nontrivial knot types 72 72 78 86 87 90 Chapter VII • The Free Calculus and the Elementary Ideals Chapter VIII 94 94 96 100 101 Introduction The group ring The free calculus The Alexander matrix The elementary ideals The Knot Polynomials Introduction The abelianized knot group The group ring of an infinite cyclic group The knot polynomials Knot types and knot polynomials Chapter IX 110 III 113 119 123 Characteristic Properties of the Knot Polynomials Introduction Operation of the trivializer Conjugation Dual presentations 134 134 136 137 Appendix I Differentiable Knots are Tame 147 Appendix II Categories and groupoids 153 Appendix III Proof of the van Kampen theorem 156 Guide to the Literature 161 Bibliography 165 Index 178 Prerequisites For an intelligent reading of this book a knowledge of the elements of 1110dern algebra and point-set topology is sufficient Specifically, we shall assume that the reader is familiar with the concept of a function (or mapping) and the attendant notions of domain, range, image, inverse image, one-one, onto, composition, restriction, and inclusion mapping; with the concepts of equivalence relation and equivalence class; with the definition and elementary properties of open set, closed set, neighborhood, closure, interior, induced topology, Cartesian product, continuous mapping, homeomorphism, eonlpactness, connectedness, open cover(ing), and the Euclidean n-dimen~ional space Rn; and with the definition and basic properties of homomorphism, automorphism, kernel, image, groups, normal subgroups, quotient groups, rings, (two-sided) ideals, permutation groups, determinants, and Inatrices These matters are dealt with in many standard textbooks 'Ve may, for example, refer the reader to A H Wallace, An Introduction to Algebraic 'Fopology (Pergamon Press, 1957), Chapters I, II, and III, and to G Birkhoff and S MacLane, A Survey of Modern Algebra, Revised Edition (The Macl}lillan Co., New York, 1953), Chapters III, §§1-3, 7,8; VI, §§4-8, 11-14; VII, ~5; X, § §1, 2; XIII, §§1-4 Sonle of these concepts are also defined in the index In Appendix I an additional requirement is a knowledge of differential and integral calculus l he usual set theoretic symbols E, c, ~, =, U, (1, and - are used For the inclusion symbol we follow the common convention: A c B means that 1) E B whenever pEA For the image and inverse image of A under f we write either fA andf -1 A, or f(A) and! -l(A) For the restriction off to A we writef A, and for the composition of two mappings!: X ~ Y and g: Y ~ Z wo write gf When several mappings connecting several sets are to be considered at the ~alne time, it is convenient to display them in a (mapping) diagram, such as l I f g X~y~Z ~1/· r I ('(l,('h eh'J)\('IlL in ('Hell ~('L di~play('d ill a dingl'atll Il:,s aL Ino~L Ol\(' illlag(' 1(,In('IIt, in allY giv('J} :.·wL of LIlt, di:l,grnJ)l, t,lle' dia,L!;ralll is ~aid t,o I ("oll8;:·d('II' PREREQUISITES Thus the first diagram is consistent if and only if gf == I andfg == I, and the second diagram is consistent if and only if bf == a and cg == b (and hence cgf == a) The reader should note the following "diagram-filling" lemma, the proof of which is straightforward If h: G -+ Hand k: G -+ K are homomorphisms and h is onto, there exists a (necessarily unique) homomorphism f: H -+ K making the diagram G H /~ f ) K consistent if and only if the kernel of h is contained in the kernel of k 168 1944 1947 1948 1949 1950 BIBLIOGRAPHY Tietze, H EIN KAPITEL TOPOLOGIE Zur Einfuhrung in die Lehre von den verknoteten Linien Teubner, Leipzig und Berlin (Hamburger Mathematische Einzelschriften 36); M R 8, 285 Ashley, C W THE ASHLEY BOOK OF KNOTS Doubleday and Co., N.Y Artin, E "Theory of braids." Ann of Math., vol 48, pp 101-126; Zbl 30, 177; M.R 8, 367 Artin, E "Braids and permutations." Ann of Math., vol 48, pp 643-649; Zbl 30, 178; M.R 9, Bohnenblust, F "The algebraical braid group." Ann of Math., vol 48, pp 127-136; Zbl 30, 178; M.R 8, 367 Borsuk, K "An example of a simple arc in space whose projection in every plane has interior points." Fund Math., vol 34, pp 272-277; Zbl 32, 314; M.R 10, 54 Borsuk, K "Sur la courbure totale des courbes fermees." Annales de la societe Polonaise de mathematique, vol 20, pp 251-265; M.R 10, 60 Chow, W L "On the algebraical braid group." Ann of Math., vol 49, pp 654-658; Zbl 33, 10; M.R 10, 98 Fox, R H "On the imbedding of polyhedra in 3-space." Ann of Math., vol 49, pp 462-470; Zbl 32, 125; M.R 10, 138 Fox, R H and Artin, E "Some wild cells and spheres in three-dimensional space." Ann of Math vol 49, pp 979-990; Zbl 33, 136; M.R 10, 317 Higman, G "A theorem on linkages." Quart J of Math (Oxford series), vol 19, pp 117-122; Zbl 30, 322; M.R 9, 606 Burger, E "Uber Schnittzahlen von Homotopie-ketten." Math Z., vol 52, pp 217-255; Zbl 33, 307; M.R 12, 43 Fary, I "Sur la courbure totale d'une courbe gauche faisant un noeud." Bulletin de la Societe Mathematique de France, vol 77, pp 128-138; M.R 11,393 Fox, R H "A remarkable simple closed curve." Ann of Math., vol 50, pp 264-265; Zbl 33, 136; M.R 11, 45 Schubert, H "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten." Sitzunysberichte der Heidelberger Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse, No.3, pp 57-104; Zbl 31, 286; M.R 11, 196 Seifert, H "Schlingknoten." Math Z., vol 52, pp 62-80; Zbl 33, 137; M.R 11, 196 Artin, E "The theory of braids." American Scientist, vol 38, pp 112-119; M.R 11, 377 Blankinship, W A and Fox, R H "Remarks on certain pathological open subsets of 3-space and their fundamental groups." Proceedings of the American Mathematical Society, vol 1, pp 618-624; Zbl 40, 259; M.R 13,57 Burger, E "Uber Gruppen mit Verschlingungen." Journal fur die Reine und Angewandte 1\;lathematik, vol 188, pp 193-200; Zbl 40, 102; M.R 13, 204 Fox, R H "On the total curvature of some tame knots." Ann of Math vol f>2, pp 258-260; Zbl :l7, :~90; M.R 12, :~73 Cf'HOllh, W ~'l)io HOlniJinoarf'n AhhiJdllngon." S.-H ll(','id(',ll)(~rger Alcad Jr iss III (1 t h N (( t Ie l" p p 2() f> 272; 1\'1 I{" I: ~, H) 2, Milnor, J \V "(hl Lhn t/oLnl c'ttr'VaLllI'C' of kllof,~." :I/UI (~lll1((lh., vol f>2, pp ~·IH ~.r~7; Zid :~7, :~HH; M.IL ~~, :\7:L BIBLIOGRAPHY 1951 1952 1953 1954 16H Seifert, H "On the homology invariants of knots." Quart J Math Oxjord, vol I, pp 23-32; Zbl 35, Ill; M.R II, 735 Seifert, H and Threlfall, W "Old and new results on knots." Canadian Journal oj Mathematics, vol 2, pp 1-15; Zbl 35, 251; M.R 11,450 Blanchfield, R C and Fox, R H "Invariants of self-linking." Ann of Math., vol 53, pp 556-564; Zbl 45, 443; M.R 12, 730 Blankinship, W A "Generalization of a construction of Antoine." Ann of Math., vol 53, pp 276-297; Zbl 42, 176; M.R 12, 730 Chen, K T "Integration in free groups." Ann of Math., vol 54, pp 147162; Zbl 45, 301; M.R 13, 105 Torres, G "Sobre las superficies orientables extensibles en nudos." Boletin de la Sociedad Matematica Mexicana, vol 8, pp 1-14; M.R l:J, 375 Chen, K T "Commutator calculus and link invariants." Proc A.M.} ') vol 3, pp 44-55; Zbl 49, 404; M.R 13, 721 Chen, K T "Isotopy invariants of links." Ann of Math., vol 56, pp 343-353; Zbl 49, 404; M.R 14, 193 Fox, R H "On the complementary domains of a certain pair of inequivalent knots." Koninklijke N ederlandse Akademie van W etenschapp(~n Proceedings, series A, vol 55 (or equivalently, I ndagationes M athemat'i(:(l,t~ ex Actis Quibus Titulis vol 14), pp 37-40; Zbl 46, 168; M.R 13, HHH Fox, R H "Recent development of knot theory at Princeton." Proceedings oj the International Congress of Mathematics, Cambridge, 19GO, vol 2, pp 453-457; Zbl 49, 130; M.R 13, 966 Moise, E E "Affine structures in 3-manifolds, V The triangulation theorem and Hauptvermutung." Ann oj Math., vol 56, pp 96-114; Zbl 48, 171;M.R.14, 72 Fox, R H "Free differential calculus, I Derivation in the free group ring." Ann of Math., vol 57, pp 547-560; Zbl 50,256 Gugenheim, V K A M "Piecewise linear isotopy and embedding of elements and spheres I, II." Proc Lond Math Soc., series 2, vol :~, pp 29-53, 129-152; Zbl 50, 179; M.R 15, 336 Kneser, M and Puppe, D "Quadratische Formen und VerschlingungHinvarianten von Knoten." Math Z., vol 58, pp 376-384; Zbl GO, :H}H; M.R 15, 100 Milnor, J "On the total curvatures of closed space curves." Math(~'lJ'#,(tll:('(t Scandinavica, vol 1, pp 289-296; Zbl 52, 384; M.R 15, 465 Plans, A "Aportaci6n al estudio de los grupos de homologia IOH 1'(\cubrimientos ciclicos ramificados correspondientes a un nudo." llwm:8ln de la Real A cademia de Ciencias Exactas, Fisicas y Naturales de lVl ad'f,;t! , vol 47, pp 161-193; Zbl 51, 146; M.R 15, 147 Schubert, H "Knoten und Vollringe." Acta Math., vol 90, pp I:n 2HH; Zbl 51, 404; M.R 17, 291 Torres, G "On the Alexander polynomial." Ann of Math., vol ri7, pp 57-89; Zbl 50, 179; M.R 14, 575 Bing, R H "Locally tame sets are taIne." Ann of Math., vol riB, pp 14fi-lfi8; Zbl fifi, InS; M.R ]fi, HI(L I~'ox, }{ H "f1'reo difforontial eulclIhlH, J r 'rho iHOrllOf'phiHrll probll'ltl." Ann (1 Malh., vol riH, pp IBH 2]0; M.lt J[), H:n Ilornrna, 'r "()n f,ho oxiHf,nrH'o or IIltIOlof,f,('d polygollH on ~·rnallif()ldH ira E:'." O,'Ulka ll;fnth('''Mlt;('a! '/ournal, vol n, pp I~B 1:\·1; Zhl [)[), {I~I; M.I t I n, I HO 170 BIBLIOGRAPHY Kyle, R H "Branched covering spaces and the quadratic forms of a link." Ann of llIath., vol 59, pp 539-548; Zbl 55, 421; M.R 15, 979 Milnor, J "Link groups." Ann of Math., vol 59, pp 177-195; Zbl 55, 169; M.R 17,70 l\1oise, E E "Affine structures in 3-manifolds VII, invariance of the knot types; local tame imbedding." Ann of Math., vol 59, pp 159-170; Zbl 55, 168; M.R 15, 889 Schubert, H "Uber eine numerische Knoteninvariante." Math Z., vol 61, pp 245-288; Zbl 58, 174; M.R 17, 292 Torres, G and Fox, R H "Dual presentations of the group of a knot." Ann of Math., vol 59, pp 211-218; Zbl 55, 168; M.R 15, 979 1955 Kyle, R H "Embeddings of Mobius bands in 3-dimensional space." Proceedings of the Royal Irish Academy, Section A, vol 57, pp 131-136; Zbl 66, 171; M.R 19, 976 Montgomery, D and Samelson, H "A theorem on fixed points of involutions in S3." Canadian J Math., vol 7, pp 208-220; Zbl 64, 177; M.R 16, 946 Papakyriakopoulos, C D "On the ends of knot groups." Ann of Math., vol 62, pp 293-299; Zbl 67, 158; M.R 19, 976 Reeve, J E "A summary of results in the topological classification of plane algebroid singularities." Rendiconti del Seminario M atematico di Torino, vol 14, pp 159-187 1956 Aumann, R J "Asphericity of alternating knots." Ann of Math., 'vol 64, pp 374-392; Zbl 78, 164; M.R 20, 453 Bing, R H "A simple closed curve that pierces no disk." Journal de Alathematiques Pures et Appliquees series 9, vol 35, pp 337-343; Zbl 70, 402; M.R 18,407 Fox, R H "Free differential calculus, III Subgroups." Ann of Math., vol 64, pp 407-419; M.R 20, 392 Schubert, H "Knoten mit zwei Briicken." Math Z., vol 65, pp 133-170; Zbl 71, 390; M.R 18, 498 1957 Bing, R H "Approximating surfaces with polyhedral ones." Ann of Math., vol 65, pp 456-483; M.R 19, 300 Blanchfield, R C "Intersection theory of manifolds with operators with applications to knot theory." Ann of Math., vol 65, pp 340-356; Zbl 80, 166; M.R 19, 53 Conner, P E "On the action of a finite group on Sn X Sn." Annals of Math., vol 66, pp 586-588; M.R 20, 453 Fox, R H "Covering spaces with singularities." Lefschetz symposium Princeton Mathematical Series, vol 12, pp 243-257; Zbl 79, 165; M.R 23, 106 Fox, R H and Milnor, J W "Singularities of 2-spheres in 4-space and equivalence of knots." Bull A M S., vol 63, p 406 Kinoshita, S and Terasaka, H "On unions of knots." Osaka Math J., vol 9, pp 131-153; Zbl 80, 170; M.R 20, 804 Milnor, J "Isotopy of links." Lefschetz symposium Princeton Math Ser., vol 12, pr 280-306; Zhl 80, 169; M.lL 19, 1070 j )apakyriakopoulos, (~ D "()n 1)nhn 'H It~rllrlla aut I Lht, UHf )hnrieity of knots " !)nw Not /I (·ad ~""l('i (I.S.A., vo! ;j:~, pp IfiB 17"2; linn (~f f\./olh., vol nn, pp I ~f); Zhl 7H, IH·t; M.IL IH, r)BO; I~}, 74)1 BIBLIOGRAPHY 171 Plans, A "Aportaci6n a la homotopia de sistemas de nudos." Revista Matematica Hispano-Americana, Series 4, vol 17, pp 224-237; M.lL 20, 803 1958 Bing, R H "Necessary and sufficient conditions that a 3-manifold be S3." Ann of Math., vol 68, pp 17-37; Zbl 81, 392; M.R 20, 325 Fox, R H "On knots whose points are fixed under a periodic transformation of the 3-sphere." Osaka Math J., vol 10, pp 31-35; Zbl 84, 395 Fox, R H "Congruence classes of knots." Osaka Math J., vol 10, pp 37-41; Zbl 84, 192 Fox, R H., Chen, K T., and Lyndon, R C "Free differential calculus, IV The quotient groups of the lower central series." Ann of Math., vol 68, pp 81-95; M.R 21, 247 Hashizume, Y "On the uniqueness of the decomposition of a link." Osaka Math J., vol 10, pp 283-300, vol II, p 249; M.R 21, 308 Hashizume, Y and Hosokawa, F "On symmetric skew unions of knotH." Proceedings of the Japan Academy, vol 34, pp 87-91; M.R 20, 804 Hosokawa, F "On V-polynomials of links." Osaka Math J., vol 10, pp 273-282; M.R 21, 308 Kinoshita, S "On vVendt's theorem of knots II." Osaka Math J., vol 10, pp 259-261 Kinoshita, S "On knots and periodic transformations." Osaka Math J., vol 10, pp 43-52; M.R 21, 434 Kinoshita, S "Alexander polynomials as isotopy invariants, I." Osaka Math J., vol 10, pp 263-271; M.R 21, 308 Kirkor, A "A remark about Cartesian division by a segment." Bulletin de l'Academie Polonaise des Sciences, vol 6, pp 379-381; M.R 20,580 Kirkor, A "Wild O-dimensional sets and the fundamental group." Jtl und Math., vol 45, pp 228-236; Zbl 80, 168; M.R 21, 300 Murasugi, K "On the genus of the alternating knot, I." Journal of the Mathematical Society of Japan, vol 10, pp 94-105; M.R 20, 1010 Murasugi, K "On the genus of the alternating knot, II." J :'l1ath k, pp 206 225 Mllr'nslIgi 1( "Nou-alnplti4, I ()2 A semigroup is a catogory !,lln.t only ono id(~nt.ity Hornilinpu,r 7:1 Hirnpln put.h Hirllplo knot 7:~ Hilll} )Jy -l'O/U""('/t't! 140 hU.H 182 INDEX single knot skew lines solid torus 55, 61 space - , covering - , Euclidean, 1, - , projective, sphere 55, 66, 161-163 splice splittable link 163 square knot standard reduction 33 stopping time 14 stevedore's knot subgroup - , commutator, 47, 108 - , fully normal, 47 - , word, 37, 47 sum, exponent, III surface 109, 162 syllable 31 symmetric group 51, 90, 92, 93 table - , group, 37 - , knot, 11, 164 - , link, 164 tame knot 5, 11, 62, Ill, 147-152, 162, 163 tame link 161 terminal point 14, 18 time, stopping, 14 Tietze equivalence 43,91,105, 106,123 Tietze theorem 37, 43ff, 44, 104, 113 toroidal neighborhood 62 torus 55, 61, 67, 132 - , double, 71 torus knot 92 total curvature transformation of finite period 163 transpose of a matrix 144 trefoil = clover leaf knot triple point 6, trivial knot trivial unit 117 triviality problem 41 trivializer 96, 134-136 true lover's knot Turk's head knot tying type - , alternating, 12 - , homotopy - , isotopy, - , knot - , presentation - , tame, 5, 11 - , trivial, 5, 11 - , wild, unbranched covering space 162 undercrossing 7, 12, 73 underlying set of generators 40 underpass 72, 73 underpresentation 72, 76ff, 134, 143 unique factorization domain 115, 162 unit 113 - , trivial, 117 untying 3,6 van Kampen theorem 54,63,65,69-71 80, 156-160 Verkettung = link Verschlingung = link vertex of a knot 5, 6, V iergefiechte 163 wild knot wildness 164 winding number 28 Wirtinger presentation 72, 86, 113 words 31 - , empty, 31 - , equivalent, 32 - , product of, 31 - , reduced, 32,33,34,35 word probleln 32,41,47 word subgroup 47,51 Zopf = braid ... would be to prolong the ends to infinity; but a simpler method is to splice them together A(~cordingly, we shall consider a knot to be a subset of 3-dimensional space whieh is homeomorphic to a circle... true equivalence relation Equivalent knots are said to be of the same type, and each equivalence class of knots is a knot type Those knots equivalent to the unknotted circle x + y2 = I, Z = 0, are... wild knots For example, no knot that lies in a plane is wild Although the study of wild knots is a corner of knot theory outside the scope of this book, Figure gives an example of a knot known to