Knot Groups. Annals of Mathematics Studies. (AM-56), Volume 56 9781400882038

Table of contents :
CONTENTS
CHAPTER I: INTRODUCTION
§1. Introduction
CHAPTER II: NOTATION AND CONVENTIONS
§1. Introduction
§2. Group Theory
§3. Geometric Conventions
CHAPTER Ill: COMBINATORIAL COVERING SPACE THEORY FOR 3-MANIFOIDS
§1. Introduction
§2. Computation of π1 from a Maximal Cave
§3. The Splitting Complex
§4. A Splitting Complex for a Knot
§5. Construction of Coverings from a Splitting Complex
CHAPTER IV: THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX
§1. Introduction
§2. An Orientable Surface Spanned by a Knot
§3. The Infinite Cyclic Covering of a Knot
§4. A Property of the Surface of Minimal Genus
§5. The Structure of the Commitator Subgroup of a Knot Group
§6. The Alexander Matrix (tentative Description)
§7. The Alexander Polynomials
§8. The Alexander Matrix
§9. The Alexander Polynomial
CHAPTER V: SUBGROUPS
§1. Introduction
§2. Kernels of Maps to Zn
§3. Kn/[Kn, Kn]
§4 . Abelian Subgroups
§5. Homology of Subgroups
§6. Commutator and Central Series
§7. Ends
CHAPTER VI: REPRESENTATIONS
§1. Introduction
§2. Metacyclic Representations
§3. Non-trivial Representations of Non-trivial Elements
§4. The Range of Finite Homomorphs
CHAPTER VII: AUTOMORPHISMS
§1. Introduction
§2. Outer Automorphisms
§3. Symmetries
CHAPTER VIII: A GROUP OF GROUPS
§1. Introduction
§2. The Semi-group of Knots
§3. Some Axioms
§4. A Binary Relation
§5. Some Examples
§6. Amalgamations
§7. Knot Groups
CHAPTER IX: THE CHARACTERIZATION PROBLEM
§1. Introduction
§2. Necessary and Sufficient Conditions
§3. Sufficient Conditions
§4. Necessary Conditions
CHAPTER X: THE STRENGTH OF THE GROUP
§1. Introduction
§2. Homotopy Type
§3. The Topological Type of the Complement of a Knot
§4. Knot Type
CHAPTER XI: PROBLEMS
§1. Introduction
§2. Problems
APPENDIX
BIBLIOGRAPHY
INDEX

Citation preview

Annals of Mathematics Studies Number 56

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse I.

Algebraic Theory of Numbers, by

3.

Consistency of the Continuum Hypothesis, by

H erm ann W ey l

11.

Introduction to Nonlinear Mechanics, by N.

16.

Transcendental Numbers, by

19. Fourier Transforms, by S.

K u r t G o d el

and N.

Kr ylo ff

B o g o l iu b o f f

C a r l L u d w ig S i e g e l

and

B o ch n er

K . C h a n d rasekh aran

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. 21. Functional Operators, Vol. I, by

24. Contributions to the Theory of Games, Vol. I, edited by H. W. 25. Contributions to Fourier Analysis, edited by A. A. P. C a l d e r o n , and S. B o c h n e r

Zygm und,

Kuhn

W.

28. Contributions to the Theory of Games, Vol. II, edited by H. W.

and A. W.

T ucker

M.

M o rse,

and A. W.

T ucker

T r an su e,

Kuhn

29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. 30.

L efsch etz

J o h n von N e u m a n n

Contributions to the Theory of Riemann Surfaces, edited by L.

34. Automata Studies, edited by C. E.

Sh a n n o n

and

et al.

A h lfo r s

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L efsc h etz

Bers,

S.

B och­

J. M c C arth y

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L efsch etz

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T ucker

K uhn

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D uncan L uce

T ucker

and A. W.

T u cker

41.

Contributions to the

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42.

Lectures on Fourier

Integrals, by S.

43.

Ramification Theoretic Methods in Algebraic Geometry, by S.

44.

Stationary Processes

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In preparation

B o ch n er.

and Prediction Theory, by H.

A bhyankar

F ursten berg

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C e s a r i,

L a Sa l l e ,

and

L efsc h etz

46.

Seminar on Transformation Groups, by

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Theory of Formal Systems, by R.

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Lectures on Modular Forms, by R. C.

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Composition Methods in Homotopy Groups of Spheres, by H.

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et al.

Sm u llya n G u n n in g

E.

N.

Steen r o d ,

T oda

written and revised by D.

E p s t e in

51.

Lectures on Morse Theory, by

J.

W.

M il n o r

52. Advances in Game Theory, edited by M. 53. Flows on Homogeneous Spaces, by L. 54. Elementary Differential Topology, by

D r esh er,

L.

Sh a p l e y ,

Auslan d er,

L.

Gr een ,

J.

R.

F.

and A.W. Hahn,

M un kres

55. Degrees of Recursive Unsolvability, by G. E.

Sa cks.

In preparation

T ucker

et al.

B.

A.

K N O T GROUPS BY

L. P. Neuwirth

PRINCETON, N E W

JERSEY

P R I N C E T O N U N I V E R S I T Y PRESS 1965

Copyright © 1965, by P r i n c e t o n U nive r s i t y P ress All Rights Reserved L. C. Card 65-14393

Printed in the United States of America by W e s t v i e w P ress ,Boulder, Colorado Princeton U

niversity

P ress O

n

D

emand

E di t i o n ,1985

CONTENTS CHAPTER Is

INTRODUCTION § 1 . Introduction................................

CHAPTER II:

NOTATION AND CONVENTIONS § i. Introduction............... .................. §2. Group Theory .................................. §3. Geometric Conventions...........................

CHAPTER Ills

CHAPTER IV:

CHAPTER V:

.

COMBINATORIAL COVERING SPACE THEORY FOR 3-MANIFOIDS § i. Introduction.................................. §2. Computation of from a Maximal Cave......... §3. The Splitting Complex........................... H. A Splitting Complex for a K n o t ................ $5. Construction of Coverings from a SplittingComplex THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX § 1. Introduction.................................. §2. An Orientable Surface Spanned by a Knot......... §3. The Infinite Cyclic Covering of a K n o t ........ §U. A Property of the Surface of Minimal Genus . . . . §5. The Structure of the CommitatorSubgroup of a Knot Group.................................. §6. The Alexander Matrix (tentativeDescription) . . . §7. The Alexander Polynomials...................... §8. The Alexander M a t r i x .......................... §9. The Alexander Polynomial....................... SUBGROUPS § 1. Introduction.................................. §2. Kernels of Maps to ZR ......................... S3. §k . §5. §6. §7.

V ..................................... Abelian Subgroups.............................. Homology of Subgroups........................... Commutator and Central Series................... Ends ..........................................

v

1

5 5 6

9 9 12

13 15 25 25 26 27 29 3U 3U U2 1*6

U9 U9 51 56 58 59 59

CONTENTS CHAPTER VI:

CHAPTER VII:

REPRESENTATIONS § 1 . Introduction................................ §2 . Metacyclic Representations.................. §3. Non-trivial Representations of Non-trivial E l e m e n t s ................................ §1*. The Range of Finite Homomorphs............. . AUTOMORPHISMS § 1 . Introduction................................ §2 . Outer Automorphisms.......................... §3. Symmetries..................................

CHAPTER VIII: A GROUP OF GROUPS § i. Introduction................................ §2 . The Semi-group of Knots...................... §3. Some Axioms.................................. §U. A Binary Relation............................ §5. Some Examples................................ §6 . Amalgamations................................ §7. Knot Groups.................................. CHAPTER IX:

CHAPTER X:

CHAPTER XI:

THE CHARACTERIZATION PROBIEM §i. Introduction................................ §2 . Necessary and Sufficient Conditions........... §3. Sufficient Conditions........................ §1*. Necessary Conditions........................ THE STRENGTH OF THE(H*OUP § 1 . Introduction................................ §2 . Homotopy Type................................ §3. The Topological Type of the Complement of a Knot...................................... §U. Knot Type. . .............................. PROBLEMS § l. Introduction................................ §2 . Problems....................................

61 61

63 6^ 67 69 72 75 76 77 78 81

83 8^ 87 87 90

91

93 93

97 99 99

APPENDIX by S. Eilenberg........................................

103

BIBLIOGRAPHY...................................................

107

INDEX

113

. .

....................................................

vi

CHAPTER I INTRODUCTION § 1 . Introduction

The relations between topology and group theory have always been cordial at the very least, and this book is meant to take advantage of that fact.

This work is intended to give some indication of the present state

of knowledge concerning the fundamental group of the complement of an arbi­ trary polygonal knot in the three sphere.

The relation of such a group to

"group theory" on the one hand, and to the problems of three-dimensional manifolds on the other is an interesting one. The homology groups of aknot group vanish in dimensions bigger then two, so that in a sense a knotgroup

is close to being free.

But,

what is more to the point, the knotgroup

arises in a natural way from a

geometrical situation, and this situation permits the application of a good many geometric techniques. and suggest algebraic techniques.

These ‘techniques lead to algebraic results, So that, just as, classically, an ab­

stract group theory arose from geometric considerations, one may hope that more sophisticated geometric attitudes will enrich group theory.

Now re­

cent results concerning three-dimensional manifolds have considerably ad­ vanced our knowledge , and the work of Wallace [67], and Lickorish [3^1, as well as the older work of Alexander [1 ], have placed knot theory in a more central location with respect to the theory of three-dimensional mani­ folds.

I hope to present the reader same of the knot theory having to do

particularly with the group of a knot.

It will be seen that the geometry

of the situation can rarely be ignored, and the interplay between algebra and geometry should be apparent. 1

I . INTRODUCTION

2

Of fundamental importance to this presentation is the concept of a covering space.

In order to present this idea coherently, I find it con­

venient to develop a theory of covering spaces of three-dimensional mani­ folds from a purely combinatorial point of view.

This theory arises from

an algorithm for computing a presentation of the fundamental group of the complement of a (possibly empty) one-dimensional subcomplex of a triangu­ lated three-dimensional manifold.

While a more general approach would have

been possible, there seemed to be no need here to consider dimensions bigger than three. One might adopt the view that a knot group is simply a peculiar special case of a group having a presentation of the following sort

(x,

V

Xj x'*J -

xj+1)

This is shown for example in [5*0.

J - 1 , 2,

. ± 1

n-i,

.

One may also derive the existence of

such a presentation from results in Chapter III.

This point of view may

occasionally prove fruitful, but I have not been able to utilize it often enough in this book to make such a position worthwhile.

It should not,

however, be ignored as a possible entree into the subject. Material which has not appeared elsewhere includes all of Chapters VIII and III, and practically all unreferenced theorems. Proofs are not given for all the theorems presented.

Proofs are

given when the methods of proof are interesting, or when the proofs have not appeared elsewhere.

Occasionally a proof has been omitted because I do

not understand it well enough to present it. The reader familiar with knot theory will find at least the following omitted:

Trotter's work on the cohomology of knot groups [63),

the fascinating results of Crowell [8 ], Murasugi (39), and Kinoshita [3i1 on alternating knots, an exposition of a good deal of Crowell's recent in­ vestigations into the structure of the commutator module [9 ), Seifert's computations concerning the Alexander matrix [57), the relations between knots and braids, a discussion of the Stallings Pibrations [59), and final­ ly links.

§1 .

INTRODUCTION

3

I thank Professors Pox, Papakyrikopoulos, Trotter, Stallings and Crowell for their generous help in the preparation of this manuscript. The invaluable assistance of Peter Strom Goldstein is also grate­ fully acknowledged. Finally, I wish to thank both Professor Gunning for his confi­ dence and cooperation, and the Institute for Defense Analyses, whose atmos­ phere and encouragement provided a great stimulus for working on this material.

CHAPTER II

NOTATION AND CONVENTIONS

§ 1.

Introduction

While most of the material in this chapter is a formality, there are some mildly unusual uses of familiar terminology.

The careful

reader is recommended to read § 3 of this chapter before reading Chapter III.

§2 . Group Theory

G subgroup.

will commonly denote a knot group and

However, [H, H]

and

H1

denote the commutator subgroup of

G ! Its commutator

will be used interchangeably to H.

Frequently reference will be made to a free product with amalgamation.

We denote such a construction A * B, and depend upon the C surrounding context to supply the information about the amalgamating maps

C -*• A,

C

B.

ample in [33].

The formal construction of A * B may be found for exC There one also may find the main properties that will be

used here. Where confusion might arise we adopt the usual notation (X,, ..., X jj j r,, ..., rm )

and

|x,, ...,

; r, , ..., r j

to distln-

guish the presentation of a group from the symbol denoting the group itself. The deficiency of a presentation is the number of generators minus the number of relations. 5

6

II.

NOTATION AND CONVENTIONS

The cyclic group of order

n

is denoted

The infinite cyclic group is denoted ^ - ♦ F - * - G - ^ Z - *' 0

When a situation G,

of the non-abelian group

F,

Z.

arises, (the extension, 1 , by the infinite

with identity

cyclic group) we will frequently refer to a generator have as if

t c G.

t

such that

a ( 5 ( a a ) ) ,

cpb ( 8 ( a b ) ) )

q>b (5 (ab )))

.

maps a to b .

This construction leads to a rule telling you which copy of

M

you walk into when you are in a copy of M and you cross a 2-simplex corre­ sponding to a 2-simplex of K.

This means we may associate to a path in M,

and a sheet M& , a path in our identification space, by walking along a copy­ ing path which starts in Ma and enters a different sheet when our assignment says to (recalling the process of analytic continuation). Now remove from this identification space all 1 -simplices which are mapped onto

L

by the various

cp& .

18

III.

COMBINATORIAL COVERING SPACE THEORY FOR 3-MANIFOILS

Having performed all the identifications and removals we were supposed to, we denote by from

M

to

M-L

M

the resulting space, and by

induced by the mappings

consider the open 3 -simplices of THEOREM 3.5.1. M. PROOF.

S

to be contained in

(M, «)

Obviously, •

1 . If

M-L,

M&

*

the mapping

q>a . In what follows we may M.

is a covering space of

is continuous and maps

M

onto

M.

is a point in the interior of a 3-simplex of

then the interior of that 3 -simplex is evenly covered by a copy of

that 3 -simplex in each copy of 2 . If

S

M.

is a point in the interior of a 2-simplex,

E,

then the (open) coboundary of that (open) 2 -simplex is a neighborhood of S

that is evenly covered.

owns a copy of M

BE,

For if

while if E

are incident along a 2-simplex

copy of

M

owns a copy of

tion assigned to

E, 3.

not contained in having covered.

T

BE,

E is not in

is in

K

S

E* mapped by

$ into

E,

S

beginning In a 3-simplex

r1

a € A

the successive images of the 2 -simplices in

s(T)

T

3 1 € A,

fixing

L

then we may denote by

, a2,

sets of the form

cr

« n+1

induced by the permutations corresponding to lying in

K,

and the ordering of the coboundary

crn+1 =

Byour condition

a . Now for each selection of an index

there is induced in this way a unique sequence

..., 8n . Furthermore, if the interior of

along a small

i^, r2, ..., r ,

of these 2 -simplices induced by our little oriented loop. on permutation assignments

M-L

and not intersecting any

i-simplices, we pass successively through 3 -simplices, If we select an index

T,

which is evenly

This may be shown as follows, if we run around L

or a single

is a point in the interior of a 1 -simplex

on their boundary is a neighborhood of

oriented loop

M

M.

then the set of all (open) 3 -simplices in

L,

of

depending upon the action of the permuta­

on the index of the copy of If

K theneach copy

either two distinct copies of

Ui_1 0,

r1

U r2

^ P1#

aj ^ 0 j •

P2,

follows from this that

is evenly covered by the interior of the disjoint 0

U *** U rnn , where

0i

Is the unique

§5.

19

CONSTRUCTION OP COVERINGS PROM A SPLITTING COMPLEX

3 -simplex in

M^

lying over k.

r^.

Suppose now that

s

Is a vertex in

M-L.

M -L

is a manifold, the star of

s

3 -ball

whose boundary is a triangulated 2 -sphere which

B

(containing

we denoted by

H,

s)

is a triangulated 3-ball.

As

the triangulation being induced by the 2 -skeleton of

Name the 3-simplices having s as a vertex ai ~ 3 -simplex A 1 in Ma lying over A 1 .

may be reached by a path on

,

H.

H

starting in

M.

A ^, Ag, ..., Am . Select a

The interior of any other 3 -simplex,

2 -simplices in

Select a smaller

Ar,

A}

in the star of

s

and intersecting only

Such a path determines a sequence of indices,

ce1, 0^ ,

as it passes from one 3 -simplex to another through 2 -simplices.

These indices are the sequence of images of

cr. determined by our permu-

tation assignment.

over

determination of A1

to

Now having selected A. aH

Ap

over

Ap

isindependent

Ap . This is true because two paths

A.,

we claim that the

ofthe pathchosen from

a,and P

from

A1

to

Ap

determine a loop

L = crp- 1

0 -skeleton of

may be deformed in the complement of the o-skeleton of

H,

H

and any loop

L

on

H not intersecting the

into a product of loops encircling each vertex of

of these loops induces the identity permutation on selection of an index

a1

and a 3 -simplex

neighborhood of a preimage of

s.

A1

A.

H

once, and each

This shows that the

determines uniquely a

By our construction, any identification

of vertices arises from an identification of some pair of 2 -simplices having these vertices on their boundary, so that the neighborhoods deter­ mined by distinct star of

s

a 1 and fixed

A1

are disjoint and the interior of the

is evenly covered by a set of neighborhoods which may be in­

dexed by the elements of

A.

This completes the proof of the theorem.

At this point a splitting complex for for two ends; first to obtain a presentation of obtain certain coverings of

M-L.

(M, L)

has been utilized

ir^M-L),

and secondly to

We will unify these two uses by means

cf the following theorem, which we must prove to show the utility of this combinatorial theory.

20

III.

COMBINATORIAL COVERING SPACE THEORY FOR 3-MANIFOLDS THEOREM 3 .5 .2 . [Fundamental theorem of Covering Space Theory; (for combinatorial 3 -manifolds with a i-dimensional subcomplex removed) .3 To every subgroup, S C ^((M-L), o) there corresponds a combinatorial covering space (M, ) there corresponds a subgroup S C ^(M-L) such that

PROOF. Theorem 3.5.1

*(*! (M)) = S.

A combinatorial covering is a classical covering by

so the second part of the theorem is true.

Let

K

be a splitting

Suppose

complex for

S C it^CM-L), o)

interior of a 3 -simplex.

Let

Now according to Theorem 3.3.1 pair of 3 -simplices in

5 (E)

is given.

S^

0

Assume

lies in the

A denote the set of left cosets of to each 2-simplex

E

of

K

cf

S,

jt^M-L)

S.

and ordered

there corresponds a generator of

On the other hand, to each generator of permutation of the cosets

(M, L).

jt^'M-L).

there corresponds a

determined by right multiplication

of a coset by that generator.

These twocorrespondences give rise to an

assignment of permutations of

A to the 2-simplices of

coboundary of each. a i-simplex in

M-L

of

S

^(M-L)

and hence leaves the cosets

fixed. Thus by Theorem 3.5.1

we may construct a covering from the

permutation assignment we have made.

Denote by

(M, )

t h i s covering.

l e t us use the same notation as before for the copies of

K,

that is,

Ma

of

S

which is S.

M

split along

a € A.

Take as base point in and located in

onto

and the ordered

is the identity because the product of the corres­

ponding generators is the identity of Sg

K

The permutation induced by a small closed loop about

Mg S

(the copy of itself).

M

the unique point

M

split along

We need to prove

K

0

lying over

0

with index the coset

^(M, 0)

is mapped by

4>

§5.

Any closed curve, closed curve

21

CONSTRUCTION OP COVERINGS FROM A SPLITTING COMPLEX

c

based at

c,

0,

based at

0

is mapped by

&

into a

and furthermore, the permutation induced by

this latter curve (as in the construction preceeding the proof of Theorem 3.5.1, a curve in S

M

fixed since we find ourselves back in

tracing the curve S

induces a permutation of

c

over

c.

A)

must leave the coset

M g when starting there and

But if the permutation induced by S.

c leaves

fixed, then of course

c

represents an element of

Thus,

Conversely if

c

is a curve representing an element of

then the permutation it induces must of course leave the cosets of fixed, so that

c

may be lifted to a closed curvfe

c

based at

0.

S, S Thus,

This completes the proof of the theorem.

Before making an amusing application of this means of construct­ ing coverings we make the following observation which may be easily veri­ fied by the reader. If K is a 2 -complex in a 3 -manifold M, and M - K is connected, then an assignment of a permutation to each 2 -simplex of K and ordering of its coboundary, satisfying the condition that the product of permuta­ tions corresponding to the non-abelian coboundary of a i-simplex in K - L is the identity, may be used to construct a covering of M - L . In other words, if we weaken the conditions for a splitting complex so that M - K is not required to be simply connected and K is not required to contain L, we still may construct coverings of M - L from usable permutation assignments. The reason that all subgroups of

M - K jt^M-L)

was required to be simply connected was so could be used to construct coverings, if

however we are willing to accept something less than that, then

M - K

needn't be simply connected. Now, suppose a 2 -manifold, T, is imbedded semi-linearly in a 3-manifold

M

in such a way that

M - T

is connected.

Then assigning the

22.

III.

COMBINATORIAL COVERING SPACE THEORY FOR 3-MANIFOLDS

permutation (1 2 ) to each 2-simplex of a 2 -sheeted covering of i-simplex in

T

M

T

will permit the construction of

since (1 2 ) ~ 1 = (1 2 ),

and each coboundary of a

contains precisely two 2-simplices.

Thus we have proved

THEOREM 5.5.3. If a 2 -manifold can be semilinearly imbedded in a 3 -manifold, M, with­ out separating M, then M has a connected two-sheeted covering. COROLLARY 5.5.^. Any 2 -manifold semi-linearly imbedded in L(2 n+i, q) (lens space) separates. PROOF. 2,

so

L(2 n+ 1 , q)

« 1 (L(2 n+i, q)) « Z2 n+1

which 1:1813 no subgroups of index

has no connected two-sheeted cover.

As a final application of splitting complexes, an irregular covering of a knot is described by means of the picture below; there are three sheets and the permutations assigned to a 2 -simplex and its ordered coboundary "top to bottom” are as indicated.

These assignments determine

all the other assignments to 2 -simplices and their ordered coboundaries.

Fig. III-i*.

§5.

CONSTRUCTION OP COVERINGS PROM A SPLITTING COMPLEX

23

Before passing on to some applications of the ideas in this chapter, we may define A. group n1

The Monodromy Group

S C «1

of a covering corresponding to a sub­

is the group of permutations of the cosets

S^

effected by right multiplication by elements of

representation of

«1

of

S

in

it is also the

as a group of permutations of the sheets determined

by mapping an element

g

mined by walking in M

n1

of

onto the permutation of thesheets deter­

along a closed curve representing

g. * The equiva­

lence of these two definitions follows from the proof of Theorem 5.5.1. B.

The Group of Covering Translations of a covering is the

group of permutations of the left -cosets of

S

induced by left multiplication by elements of of

S

Ma

which define a homeamorphlsm of

in

N(S).

in its normalizer, N(S)

N(S),

of these left cosets

It is also the group of those permutations of the sheets M.

Reiterating, the group of covering translations is on the one hand the normalizer of permutations of the

S

modulo

S,

and on the other the group of all

sheets which define homeomorphisms of

from the fact that a permutation of the sheets of phism of if

Mg

M

is mapped onto

Mgg

then

Mgw

is known> for where h

is mapped onto

is

for any

so that the permutation defining the home omorphi sm is induced by

left multiplication of the left cosets of

g's

plication of only when

Mg

is mapped onto

(M). This implies

multiplying any left coset of but these

defining a homeomor-

is completely determined when the image of

a generator of w €

M

M. This follows

are S

g € S,

S

S,

by elements

just the members of the normalizer of

by an element

g

g,

which when

on the left, give another left coset;

leaves the coset

S

S. Since multi­

fixed when and

the equivalence of the two definitions of the group of

covering translations follows.

CHAPTER IV

THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX

§ i. Introduction

In this chapter we shall first study the covering of the comple­ ment of a knot corresponding to the commutator subgroup of its group.

This

will lead in a natural way to the Alexander matrix. Alexander's duality theorem implies that a knot group, modulo its commutator subgroup is infinite cyclic. III, the presentation for

G

G,

As remarked in Chapter

we have obtained has deficiency

+ 1 . This

state of affairs simplifies the study of the Alexander Matrix of

G.

We

shall discuss this matrix, as well as its elementary divisors, and it will be seen that the matrix determines group.

G

modulo its second commutator sub­

The determinant of this matrix is the Alexander Polynomial and it

will also be examined.

We begin however, by proving a theorem which re­

veals the structure of the commutator subgroup of a knot group.

Later in

the chapter, Crowell's work on the abelianized commutator subgroup will be described.

§2 . An Orientable Surface Spanned by a Knot Let

M

borhood + of a knot

denote the closure of the complement of a regular neigh­ k C si

Then

M

is a manifold with boundary, and by

Alexander Duality there exists an isomorphism

f

In the sense of [6 8 ], 25

3 = k.

k.

may be found in [5 7 ] and [1 8 ].

The Infinite Cyclic Covering of a Knot denote an orientable surface contained in

Suppose that the genus of

spanning of

S

k

k

is

g,

S

g

such that

arid that any other surface

has genus greater than or equal to

Obviously As

S

S3

g.

g is called the genus

is a knot type invariant.

isorientable we may orient each 2 -simplex

in

S

and

assign an ordering to the pair of 3 -simplices in each coboundary of a 2-simplex in

S,

so that the induced orientation cf the coboundary of

these simplices agrees with some pre-assigned orientation of

S3. If

a

§3.

THE INFINITE CYCLIC COVERING OF A KNOT

is a 2-simplex, denote this ordering the integers

(n-*n+l)

6(a).

to each pair

Now, assign the permutation of

(a, 6(a)),

then this assignment

defines, via the construction in Chapter III, a covering Any element

a 6 it^S3- k)

27

M

of

S3- k.

having linking number (58)

o

with

k belongs to the subgroup of this covering, for it will have intersection number [58]

o

with

S,

and this implies that the permutation of the in­

tegers (sheets) induced by

ct is trivial.

Conversely, if a

the subgroup of this covering, it follows that o

with

k.

belongs to

must have linking number

Thus we have constructed the covering of

to the commutator subgroup of

a

S3- k

Corresponding

^ ( S 3-^.

Particularizing the description in Chapter III, we obtain by taking a countable number of disjoint copies of indexed by the integers, and pasting copy "right hand" copy of S

in copy

S, iS1

in

S3

xi-, i-is i

X = (S3 split along S),

to copy

Xi+1

along the

X^^ and the "left hand" copy

X^+1. The common copy of Copies of

Xi

k

in each

split along

M

X^

i+1S2,

of

is then removed.

S

X i+1

xi iS2

iS, i+1S2

We shall use the notation of the above picture in the next section. The group of covering translations of since the commutator subgroup is normal.

M

is infinite cyclic

The group of covering translations

may be generated by the permutation of the sheets sending X^ Denote the covering translation sending

§U.

to

Xi+1«

by tn .

A Property of the Surface of Minimal Genus

(S3 split along which we denote

X^

to

S) = X

has on its boundary two copies of

S1, S2, noting that

S1

U S2 = dX,

S1 n S2 = k.

S,

28

IV.

THE -COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX I£MMA !♦..!»..i. The inclusion map induces a monomorphism hf:

PROOF.

Since

zero, hence no power of homotopic.

k

is knotted,

k

S1 dX -* ^(dX).

is of genus greater than

considered as an element of

dX = S 1 U Sg, and

Since

S^

h:

S 1 n Sg = k,

^(S^)

is null

a simple application

of the van Kampen Theorem gives

*.(dX) * * (S.)

1 jc1 (S1)

Thus,

*

*,(SJ

1 1 *,(k) 1 2

is imbedded monomorphically in

by the inclusion

n^dX)

by a map induced

h.

LEMMA k . k . 2 . The inclusion map v: S. -♦ X JL 1 induces a monomorphism vT : (S1) -► n 1 (X) . PROOF. curve on

S1,

Lemma if.U.i.

Suppose the lemma is false, and that

such that a £ o

polygonal so that

on

X

or s o dX.

and

in X,

Since S 1

a £ o

on

a

is a closed

S 1.t According to

is polyhedral,

a

may be assumed

oc satisfy the hypothesis of the LoopTheorem

([5 0 ], Theorem 1 5 . 1 and Theorem

1 ), thus we may assume that

According to Dehn's Lemma [5 1 ],at

bounds a

in

X.If we cut

S1

along a, and sew discs to both sides ofthe

obtain a new surface rates

S1,

of

than

S.J, of

£

0

on

S1,

at

does not separate

S1, X(S1), with that of

one edge, and two faces,

curve

at

the

k.

If the curve

new surface

Sj

S1,

x(S’) - x(S.,) * 2 ,

sepa­

has lower S1

then compare the Euler Charac­

hence

so we again arrive at a contradiction.

S^

has lower genus

The existence of the

thus leads to a contradiction, so the lemma is proved.

denotes homotopic to."

cut we

Sj. Since the cut adds one vertex

REMARK. Lemmas 1 and 2 obviously remain valid if S2 is substituted for S1. Lemma 2 also remains valid if k is removed from X. s

at

S1, which contradicts the assumption that the genus of

is minimal. If teristic

S ’ which is bounded by

then because

genus than

is simple

at

non-singular polyhedral disc

§5.

THE STRUCTURE OP THE COMMUTATOR SUBGROUP OP A KNOT GROUP

29

Roughly speaking, what this lemma means is that non-contractible curves on a surface in

S3- S

S,

of minimal genus spanning

k

are non-contractible

if they are pushed off the surface (to either side).

§ 5 . The Structure of the Commutator Subgroup of a Knot Group. For the rest of this chapter

G will denote

i^CS - k).

THEOREM 4 .5 .1 . If CG, G) is finitely gen­ erated, it is free of rank 2g, where g is the genus of k. If [G, G] is not finitely generated, then either it is: A) a non-trivial free product with amal­ gamation on a free group of rank 2g, and may be written in the form

P

* A * A * A * A * A ••• PP P P P 2g 2g 2g *2g 2g *2g

,

where F2g Is free of rank 2g, and the amalgamations are all proper and identical, or B) locally free, and a direct limit of free groups of rank 2g.

REMARK. PROOF. identification of every

I do not know if case B) can occur.*

By virtue of Lemma 4.2, the last remark in §4 and the iS1

and

i+1S2,

i,

f i1! where

the following diagram is valid for

the

f i+1 2

are mon o m o r p h i s m s .

R. Crowell has informed me that E. Brown and he have proved that case B) cannot occur.

30

IV.

THE C0MUTAT0R SUBGROUP AND THE ALEXANDER MATRIX

By a simple application of the van Kampen Theorem, the funda­ mental group of

U

X^

is the direct limit of the above system.

This

* 1 (Xi) * *i(Xi+i)-

direct limit is a free product with amalgamated subgroup, Let

*i(is i>

In -Xo UX, U ••• V i

U*n.

n>0

T.n -X., U X . a U ••• X^, U X _ n, 00

Y

=

n>l

00

U x. i=i

=

Y

1

U

"

X,

i»-i

1

.

Using the fact that each factor in a free product with amalgam­ ation is contained as a subgroup in the free product with amalgamation [3 3 ] and proceeding inductively it is clear that frcm the above diagram one obtains, (I)

., 0 isomorphism of

-

([33], p. 3 2 ).

§5.

31

THE STRUCTURE OP THE COMMUTATOR SUBGROUP OP A KNOT GROUP It follows then that

^(Yj

C xJYJ,

« 1 (Y_1) C ^ ( Y ^ J .

and

This fact and the diagram above imply that

* ,0 0 = * ,o r _ . U Y J = « ,( Y .J Note that if

k is of genus

g, then

1=1.

Then no

jf^

« ,( T J ^(S)

.

is free of rank

x J q S J -► « 1 (XQ)

Suppose one of the maps, say for

.

2 g.

is not onto,

will be onto, so that

U Y J c «,(y_ 8 U y J

*,0rj c «,(y_,

•••

and

*,(X) so that is onto

xJX)

«t(S)

* ^(XJ

then all

- ^(YJ

* V Y -n U YJ

and

« 1 (X)

is free of rank Qf 1

generated,

«,(Y.n U Y j *1 (0 S1) -*■ *.(X0)

is not finitely generated. But if

i = i, 2 ,

for

U

n= 1

and

Qf2

jf^

are onto so that

** x j Y j - ^ ( Y J ~

^ *i(Y-oo U YJ

2 g.

Hence if

are onto and

(Y_ 1 U Y J

- *i(X>

[G, G] = x^X)

x^X)

is finitely

is free of rank

2 g.

This

proves the first assertion of Theorem 4A.1.

If neither of the mappings jt.(Y 1

) * x,(S)

x (Y ) 1

is a proper free

),

free product with amalgamation of 2g

is proven.

is onto, then

wjX) =

productwith amalgamation and may

00

be written as ii5L> x (Y U Y n > o 11

of rank

Q? 1

isomorphic to

where each

^(YJ

xJS),

Suppose one of the maps,

so

x (Y U Y ) > n -n

and

is aproper

on a free S1*011?

by virtue of equations i and 2 , A) Qf^,

is onto,

say

Qf2, and the

32

IV.

other,

THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX

is not, then % *i o 00

U Y _)

• Since the

mappings in this direct system are allinclusions, it follows finitely generated subgroup H

H C ^(X)

that any

lies in some ^(Y^ U Y_n) and so

is a subgroup of a free group and hence free [U7]; in other words,

jt^X) = [G, G]

is locally free, and

proof of the theorem.*

B)

is proven.

This completes the

This theorem has many implications which will be

found scattered throughout the remainder of the book.

We state one corol­

lary here which will be used in this chapter. COROLLARY U.5 .1 . The center of trivial. PROOF. > 1

If

[G, G]

[G, G]

is

is- finitely generated, it is free of rank

according to Theorem ^.5 .1 , hence If

[G, G]

is not finitely

the corollaryfollows. generated, itis either a free

product with amalgamation on a centerless group and hence centerless according to [3 3 ] p. 3 2 , or else locally free and non-abelian and hence centerless. Q.E.D. * The homeomorphism may be taken to be the covering translation restricted to Y -n _ U Y °°. * The proof given above is identical to that given in [^U].

tn

§5.

THE STRUCTURE OP THE COMMUTATOR SUBGROUP OP A KNOT GROUP Whether cr net

upon

G*

33

is finitely generated we may still look

G ? as in the picture below

A generator, acts upon

t,

G ! by mapping

duced by the action of Thus

G

t

t

A^

upon

G/G1 of covering translations

1 by the natural isomorphism in­

on

may be described by giving the structure of

indicated in Theorem 4.5.1, jugation by

of the group

since the automorphism of

coincides with the action of

t

on

G*

as

G 1 induced by con­

G T induced by a

covering translation. It is appropriate to mention here two results of Crowell [9 ] and Rapaport C53 3, and a result of Murasugi [4ol.

First, a theorem which

will be proved in § 9 of this chapter: THEOREM 4 .5 .2 . (Rapaport-Crowell) If G ’ is finitely generated, then |Aq(o)| = 1 . Here

A^(t)

is the Alexander polynomial, an invariant to be

described in §7. Secondly, a partial converse to Theorem 4.4.1, THEOREM 4.5.3. (Rapaport-Crowell) If is free it is finitely generated.

G1

Finally a theorem giving sufficient conditions for finitely generated. THEOREM 4 .5 .4 . (Murasugi) If k is alternating and |a(o)| = 1 , then G* is finitely generated.

G*

to be

IV. §6.

THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX

The Alexander Matrix (Tentative Description)

In a paper [2] published in 1933, J. W. Alexander constructed over a certain ring, a matrix whose determinant, he proved, was up to a unit an invariant of the knot type of

k.

Since that paper, the matrix

he constructed has been constructed by several other people by several different methods [20], (57], [M ]. We shall give an interpretation of the Alexander matrix in this section, but the proof will be postponed. Recalling Chapter II, 1 - G'/G" - G /G" - G/G’ a module over

o

A)

|t: |, on

G'/G".

G'/G", since

G/G’ is free, thus

G/G’. It is clear that the structure of

with the module structure of

® G'/G"

G/G* = |t: | acts on

isexact and

G/G"

is

is tied up

We now assume:

The Alexander Matrix of

k

is a presentation

[9 ] of

considered as a module over the integral group ring of where

G'/G"

G/G’ acts by multiplication on

G/G'

G/G"

G/G’ =

and by conjugation

G ’/G”. It will also be convenient to assume the following easily proved

fact. B)

Suppose

G, H

areknot groups, then

G ’/G"

isomorphic as modules over the integral group ring of respectively if and only if

G/G"

and

H/H"

and

G/G'

H'/H11

are isomorphic.

We remark finally that the Alexander Matrix of a knot group may be computed from a finite presentation of

§7. If then the ideal, of

G.+

A

is an a,

are

and H/H'

G

G,

[2 0 ].

The Alexander Polynomials n columned Alexander Matrix for a knot group

generated by the minors of rank

n-d

G,

is an invariant

A generator of the smallest principal ideal containing

a

is

* It is proved in Zassenhaus, Theory of Groups, that given a matrix, these ideals are invariants of a module which the matrix may be assumed to present, and we have assumed that the Alexander Matrix, in fact, presents a module which is an invariant of the knot group. Accepting these facts makes most of this section academic.

§7. called the

THE ALEXANDER POLYNOMIALS

dth Alexander Polynomial of

unit, of course.

G;

35

it is determined only up to a

The invariance of these polynomials is not obvious, nor

is it trivial to prove.

One proof depends upon the Tietze Theorem on

equivalence of presentations of isomorphic groups [2 0 ].

To understand this

proof one should appreciate the manner in which the Alexander Matrix may be constructed from a finite presentation of a group.

We do not wish to de­

velop the free calculus here, so that we must refer the reader to [2 0 ] and [1 9 ] for a full account of the construction of the Alexander Matrix.

The

proof goes roughly as follows: 1 . Denote by

G, H

two knot groups, by

the infinite cyclic group generated by 2.

Suppose

3.

There must exist

G

and H

the group ring of

t.

are isomorphic. a finitesequence of Tietze transformations

[61] from a finite presentation of k.

JZ

G

to a finite presentation of

H.

Tietze transformation I adjoins a new relation which is a

consequence of the old relations.

The effect on the Alexander Matrix of

such an operation is to adjoin a new row which is a linear combination of old rows with coefficients units in

JZ.

This operation and its inverse

leave invariant the ideal generated by the minors of rank 5.

n-d.

Tietze transformation II adjoins a new generator and a new

relation setting this generator equal to some word in the old generators. The effect on the Alexander Matrix is to adjoin a new row, and a new column of zeros, except for the entry common to the new row and column, which will be a

1 . Obviously, the ideal generated by the minors of rank

n-d

of

such a matrix is unaffected by such an operation, or its inverse. 6. of

G

Numbers 3,

are an invariant of

and 5, imply that the Alexander Polynomials G.

36

IV.

THE COMMUTATOR SUBGROUP AND THE ALEXANDER MATRIX

We shall subsequently indicate heuristically, how the Alexander Matrix is a presentation [9 ] of

G'/Cr"

as a module over

G/G1. If as

agreed earlier we assume this last statement, an alternative proof of the invariance of the Alexander Polynomials may be given along the following lines.

We use the same notation as the previously outlined proof.* 1 . G'/Gr" = M

over thegroup

ring of

2 . If

of

M and

A

and

H f/H" = N

must be isomorphic as modules

G/Gf«* H/H1 which we denote

and

B

N respectively,

JZ.

are presentation matrices (Alexander Matrices) then the matrix, C,

below presents M

or

N.

A, B have at least as many rows as columns. (Rows of zeros must be added if necessary.) A O

I

P

I

0

B

0 C =

I Q

is the identity matrix of the appropriate size. is a matrix of zeroes of the appropriate size.

P

Q

describes the generators of the presentation of N in terms of the generators of M. describes the generators of a presentation of M in terms of the generators of N.

0

B

1

Q

3. Each row in the matrix A

0

P

I

the rows of

,

is a linear combination of

, and conversely.

Number 3 implies that the ideal generated by the minors of C of rank

n - d

is equal to the ideal generated by the minors of rank

O B n - d

of

A O or

I Q

P I

§7.

THE ALEXANDER POLYNOMIALS

ideal generated by minors of rank r + s - d of

So that

r 0

s B

1

Q

ideal generated by minors of rank u - d of

and

A

37

ideal generated by minors of rank u + v - d of ideal generated by minors of rank s - d of

=

U

V

A

0

P

I

B,

and this implies the desired result.+ If

A

denotes a square matrix presenting

G'/G",

then we

contend: PROPOSITION k .7.1. Det A every element of G'/G". PROOF. matrix

A

F/ im A

[9 ].

Denoteby x € F,

q> the natural map from

then

det A

so that

F to

xA is in the kernel of

= (adj A)A = (det A)I, we see that

x(adj A)A = yA, that

With the same notation as before, we may consider the

as defining an endomorphism of a free module

If A(adj A)

annihilates

,

which' implies

M.

In a moment a much more general theorem of Crowell will be proved. We shall postpone until § 9 some theorems about the first Alex­ ander Polynomial.

In the next section we return again to consideration of

the Alexander Matrix.

At this point we present some of the high points of

Crowell's investigations [93, [1 0 ], into the module structure of

G'/G".

The proofs we give are Crowell's. We begin by fixing notation. ring of ideal in

G ’/G".

A

denotes the

Jt' which annihilates

Jt

Jt module A.

denotes the integral group G ’/G".

, f/ Q « ^ % «p '0 (0 2' 1'0 1 =
1

so

\|r(t)

d

so that

q>(xi)

must equal *(t) = t.

d q>“1(xi)d~1= y ( X ] )

modulo ^(t)

If

freely generate i.

G ’ and

On the other hand,

♦

is non-trivial on

9 (d)xi«p(d”1) = cp2(xi)

M G”

2g,

G 1 is free of

d - F) c A,

then

(P, cppi) € A.

There is a dis­

and there is at least one

A

to

78 Ax. i*-.

VIII. If

(F, H)

* q>p(G) *

We remark thatas

are in

A,

H, q O

then is in

F and

H

q>F

or

q>g,

.

are canonically subgroups of the

free product with amalgamation, the placement of defined by either

A

p ) o (H, q>H) . Define a binary relation,

r,

between elements in

A

(F, (pF )r(H, q>H),if and only

if

¥( (F, q>p) o (H, cpH t))

as

follows:

admits an involution which induces CLAIM: PROOF.

r

i on the subgroup

Is symmetric.

Consider

F

*

t(g') = g,

H = L, and

0.

Take

G

to be the two point space

(m = n * 0),

t

to be the

interchange of these two points, and as imbeddings the maps putting these points (in either order) at the place indicated by the arrows in the above figure.

It can be seen that two whiskers

(P, q>F),

(H, F )

onto the integer (number of whiskers emanating from the image of 1) minus (number of whiskers emanating from the image of 2). Another topological adaptation of the construction is to take for the operation

o,

the cartesian product of spaces with base points, with

one subspace identified with another. and ((Y, yQ), qpy) (G, ga)

More precisely, if

are two spaces with basepoints and

((X, xQ), cp^

q>x ,

in each, then define

((X, x0), ¥x) o ((Y, y0->, * ) = ((Z, z0), (jj

where

imbed a space

$5.

SOME EXAMPLES

83

Z . X x Y/(p) r (F,

ip)

This may beproved as follows:

y ((F,

q>p) o (F, ip O ) = L

y ((F,

q>p) o (F,

is isomorphic to

by a mapping, say

p.

A meridian

m'

u

u

n (k ,, ... Kgn ; I l t K g ^ , i-1

n (H,, ... H2n: II 1 % . , , ^ l ) i=1

u

u (H g ,

...

H2 n :)

.

(K g ,

...,

Kgn )

W l ~ Yi

p:

= *n

morphism of the free group

for

G ’?

that there exist knots for which automorphism.

G*

n > 1

and

t

cp of

an auto­

Giffen proved in his thesis q> = *n

modulo an inner

(This is related to the Smith conjecture

(Chapter III) as it applies to knots with

G* finitely

generated.) K.

Can Case B) of Theorem U.5.1 actually occur?*

L.

Is the decomposition in Case A) of Theorem U.5.1unique the sense that the rank of the amalgamating subgroup is always twice the genus?

See the first footnote to Chapter IV 55.

in

§2. M.

PROBIEMS

101

Does there exist a knot group with a non-trivial symmetry leaving a peripheral subgroup (Chapter VII) element-wise fixed?

(This is sort of an algebraic version of the Smith

problem.) N.

Can a knot group be ordered? G1

with

(This is easy for knot groups

free.)

0.

What can be said of the Frattini subgroup of a knot group?

P.

Is there a simple condition on an automorphism free group

F,

of rank

is left fixed by erators of

F.

of a g ni = 1 [ai,b13

are a set of free gen­

The interest in this question of course,

stems from the hypothesis of Theorem 9 .2 .3 . Q.

Does the commutator subgroup of every knot group have cohomological dimension 1 ?

(This would provide an example of

a group with geometric dimension 2 , category 2 , and cohomo­ logy dimension 1 . Perhaps in any case a perfect commutator subgroup (see Chapter IV) of a knot group is an example of such a group). R.

Does there exist a useful algebraic theory suggested by the Morse theory in dimension three?

This might involve a gen­

eralization of Stallings* theorem (59). S.

Can a knot group contain an element equation of

T.

xn = g

g ^ 1

such that the

has solutions for arbitrarily large values

n?

Every knot group contains the group

(a, b; [a, b)).

This

subgroup may be obtained from zhe natural inclusion of the fundamental group of a non-singular torus in the knot group. Suppose a knot contains the group of a closed surface of genus

g.

of genus

Does there exist a non-singular colsed surface g

whose fundamental group is injected monomorphic

102

XI.

PROBLEMS

ally Into the knot group by the natural Inclusion? U.

Suppose

H

is a group satisfying

a) H/H* - ZR ; b) H

may be finitely presented with atleast

asmany

relations as generators. Is V.

H a

hctncmorph of a knot group

An arbitrary knot group

G ?

G must contain

a) A free group of any rank; b) A free abelian group of rank 2. Must

G

contain any other groups?

(By Theorem U.5 .1 , G W.

contains a free group

of rank 2 .)

Can the word problem be solved in a knot group? (Perhaps this follows from Theorem U.5 . 1 or a positive solution to Problem B.)

X. Y.

Can one decide algebraically if a knotgroup

is cyclic?

Can one select geometrically significantrepresentatives from each conjugacy class in a knot group?

Z.

Can the crookedness of a knot type [U9 ] be algebraically determined?

APPENDIX by S. Eileriberg

Let

A

be a category (which for the sake of simplicity will

be assumed to be small, i.e., the objects of set).

Let

6 . [21*]

H. Gluck, The Reducibility of Imbedding Problems, The Topology of 3-Manifolds and Related Topics. Prentice-Hall,1 9 6 1 .

[2 5 ] H. Freudenthal, Ueber die Enden topologischer Raume und Gruppen, Math. Zeit., 31 0931), pp. 692 -7l3. [26]

C. H. Giffen, Princeton Ph.D. Thesis, 1 9 6 ^.

[2 7 ] A. Haefliger, Knotted Ok - i )-Spheres in 6 k-Space, Ann. of Math., vol. 75, no. 3 (1 9 6 2 ). [28]

M. Hirsch and L. Neuwirth, On Piecewise Regular n-Knots, Ann. of Math. (to appear).

[2 9 ] H. Hopf, Enden offener Raume und unendliche dlskontinuierliche Gruppen, Comment. Math. Helv., _i_6 09^*0, pp. 8 1 - 1 0 0 . [30]

K. Iwasawa, Einige Satze Uber freie Gruppen, Proc. Japan Acad., V9

[31]

S. Kinoshita, Alexander polynomials as Isotopy

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INDEX Imbedding Theorem, 22

Abhyankar's Question, 61, 63 Alexander Matrix, 3h ff.; Polynomials, 3k, k6 , 86 algebraic symmetry, 73 algorithm, 87 amalgamation, free product with, 5 Annihilator Theorem, ho

longitude, 68 meridian, 68 Monodr any Group, 23 path lifting, 1 7 periodicity theorem, 53 presentation, 2, 5 Dehn, 1 h Wirtinger 1 5

cave, 9 , 10 center, 3 2 , U6 , 57, 58 coboundary, 7 ; non-abelian, 1 1 , 1 6 , 17 commutator subgroup, 2 9 , 3 3 , k7 covering, combinatorial 9 ff. covering translations, group of, 23

Smith Problem, k9 spanning surface, 2 5 - 2 5 splitting complex, 1 2 , 1.3 Structure Theorem (for G f), 29 surface, spanning, 2 5 - 2 6 symmetry, algebraic, 73

deficiency, 5 , 1 5 Duality Theorem, 91 genus, 26 Goldstein, Peter Strom, 3 group system, 67, 9k, 97

whisker space, 82

113