C-wild knots
by
Yunlin He
A thesis submitted to the Graduate Faculty of
Auburn University
in partial ful?llment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
May 5, 2013
Keywords: In?nite connected sum of knots, Wilder knots, C-wild knots
Copyright 2013 by Yunlin He
Approved by
Krystyna Kuperberg, Chair, Professor of Mathematics
Wlodzimierz Kuperberg, Professor of Mathematics
Jerzy Szulga, Professor of Mathematics
Abstract
Here we give a de?nition of in?nite connected sum of tame knots and de?ne a C-wild
knot to be an in?nite connected sum of tame knots whose wild points form a Cantor set.
We further give a classi?cation of C-wild knots in terms of Wilder knots, which are in?nite
connected sums of tame knots with one wild point.
ii
Acknowledgments
I want to thank Dr. K. Kuperberg, Dr. W. Kuperberg, and Dr. J. Szulga for their help
and support.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Tameness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.5 Some theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.6 Connected Sum of Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.7 Knot Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Generalizing Some Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 In?nite Connected Sum of Knots . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 The Wilder Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1 Wilder Connected Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 The Wilder Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 C-wild Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1 Cantor Connected Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 C-wild knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 Wilder Knots in C-wild Knots . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iv
List of Figures
4.1 Wilder knot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1 Cantor connected sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
v
Chapter 1
Introduction
In [5], Fox and Harrold gave a complete classi?cation of the Wilder arcs, which were
?rst considered by R.L. Wilder. A Wilder knot is a wild knot with exactly one wild point
and can be thought as obtained by identifying the end points of a Wilder arc. Here we
consider Wilder knots as an in?nite connected sum of tame knots, and show that if doing
in?nite connected sum in a di?erent way, we can get a wild knot whose wild points form a
Cantor set. We call such wild knots C-wild knots and give a classi?cation of these knots in
terms of Wilder knots.
In chapter 2, we presented the preliminaries. Starting from section 2.6, all knots are
assumed to be oriented and in oriented S3. In chapter 3, we generalized the concept of
connected sum of knots, and in?nite connected sum of knots is de?ned. In chapter 4,
we de?ned Wilder connected sum of knots, a speci?c way of doing in?nite connected sum,
considered Wilder Knots as the Wilder connected sum of tame knots, and gave a classi?cation
of Wilder knots. In chapter 5, we de?ned a C-wild knot to be an in?nite connected sum
of tame knots whose wild points form a Cantor set. Earlier than this, we de?ned Cantor
connected sum of knots, another way of doing in?nite connected sum. We showed that every
C-wild knot can be obtained by doing Cantor connected sum of tame knots, and gave a
classi?cation of C-wild knots based on this.
1
Chapter 2
Preliminaries
De nition 2.0.1. Rn = {x = (x1,...,xn)} = the Euclidean space of real n-tuples with the
usual norm |x| = (?xi2)1=2 and metric d(x,y) = |x?y|.
Bn = the unit n-ball of Rn de?ned by |x| ? 1.
Sn = ?Bn+1, the unit n-sphere |x| = 1.
I = [0,1] the unit interval of R1.
2.1 Orientation
De nition 2.1.1. A closed (compact, without boundary)connected n-manifold M is ori-
entable if its nth singular homology group with Z coe?cients Hn(M) = Z. If the connected
compact manifold M has nonempty boundary, it is orientable if Hn(M,?M) = Z. A choice
of one of the two possible generators of Hn(M) resp. Hn(M,?M) is called an orientation,
and an orientable manifold together with such a choice is said to be an oriented manifold.
Lemma 2.1.2. By restriction any submanifold N (n-dimensional with boundary) of an ori-
ented n-manifold M is oriented. Furthermore, the boundary ?N of an oriented n-manifold N
is oriented by choosing the (n-1)-cycle which is the boundary of the preferred relative n-cycle.
De nition 2.1.3. Let M, N be oriented n-manifolds. A homeomorphism f : Mn ? Nn is
said to preserve or reverse orientation, according as the induced homomorphism on the
n-th homology carries the preferred generator for M to the preferred generator for N, or to
its negative.
Lemma 2.1.4. Let M, N be oriented n-manifolds with boundary. Then any homeomor-
phism from M to N that is an extension of an orientation-preserving (orientation-reversing)
2
homeomorphism from a component of the boundary of M to a component of the boundary of
N is orientation-preserving (orientation-reversing).
Lemma 2.1.5. Hn(Sn) = Z; Hn(Bn,?Bn) = Z, for n ? 1.
2.2 Triangulation
De nition 2.2.1. A (Euclidean) complex is a collection K of simplexes in Rn, such that
(1). K contains all faces of all elements of K. (2)If ?,? ? K, and ??? ?= ?, then ???
is a face both of ? and of ?. (3)Every ? in K lies in an open set U which intersects only a
?nite number of elements of K.
De nition 2.2.2. Let K be a complex. A subset X of |K| is a polyhedron if there is a
subdivision K? of K and X = |S| for some subcomplex S of K?.
De nition 2.2.3. A set X is triangulable if there is a complex K such that X and |K|
are homeomorphic. K is called a triangulation of X.
De nition 2.2.4. For n ? 3, a piecewise linear manifold or PL manifold is an n-
manifold M with a ?xed triangulation. Let K be a ?xed triangulation of M, then PL M is
|K|.
The following theorems are due to E.E.Moise, see [9] for proofs.
Theorem 2.2.5. Every triangulated 3-manifold is a combinatorial 3-manifold.
Theorem 2.2.6 (The triangulation theorem for 3-manifolds). Every 3-manifold can be tri-
angulated.
Theorem 2.2.7 (The Hauptvermutung for 3-manifolds). Let K1 and K2 be triangulations
of a 3-manifold M. Then there is a subdivision K?i of Ki for i = 1,2 and a simplicial
homeomorphism ? : |K?1| ? |K?2|.
3
2.3 Knot
De nition 2.3.1. A subset K of S3 is a knot if K is homeomorphic with S1; a subset A
of S3 is an arc if it is homeomorphic with the unit interval [0,1].
De nition 2.3.2. Two oriented knots K and K? of oriented S3 are equivalent if there is
an orientation-preserving homeomorphism of S3 onto itself that takes K onto K? preserving
the orientation.
Remark 2.3.3. Depending on the context, by a knot we may also mean an equivalence class
of knots.
De nition 2.3.4. A unknot,or trivial knot is a knot equivalent to S1.
De nition 2.3.5. A polygonal knot (arc) in PL S3 is a knot (arc) that is a polyhedron.
2.4 Tameness
De nition 2.4.1. Let K be a complex. A subset X of |K| is tame if there is a homeomor-
phism h : |K| ? |K| such that h(X) is a polyhedron.
De nition 2.4.2. Let K be a complex. A subset X of |K| is locally tame at a point p of
X if there is a neighborhood N of p and a homeomorphism hp of Cl(N) onto a polyhedron
such that hp(Cl(N)?X) is a subpolyhedron.
De nition 2.4.3. A set X is tame in a topological 3-manifold M if M has a triangulation
K relative to which X is tame, it then follows that X is tame relative to every triangulation
of M by theorem 2.2.7. Otherwise, it is wild.
De nition 2.4.4. Similarly, a subset X of a 3-manifold M is locally tame at a point
p of X if M has a triangulation K relative to which X is locally tame at p. If X is locally
tame at each point of X, then it is locally tame.
De nition 2.4.5. A point p of X is called a wild point of X if X is not locally tame at p.
4
Theorem 2.4.6. In a 3-manifold, every locally tame set is tame.
Proof. By theorem 2.2.6, every 3-manifold is triangulable. By [1], every locally tame set is
tame in a PL 3-manifold. Then use theorem 2.2.7.
2.5 Some theorems
Theorem 2.5.1 (Generalized Jordan curve theorem). If S is homeomorphic to Sn?1 in Sn,
then Sn ?S has two components, and S is the boundary of each.
Proof. See [6].
Theorem 2.5.2 (PL Shoen?ies theorem). If S is a PL 2-sphere embedded in PL S3, then
the closure of the complementary components of S are PL 3-cells.
Proof. See [2] or [9].
Theorem 2.5.3 (PL annulus theorem). If B1 and B2 are PL n-cells in PL Sn, with B1 ?
Int(B2), then Cl(B2 ?B1) is PL homeomorphic to ?B1 ?I.
Proof. See [7].
Lemma 2.5.4. If C, D are homeomorphic to Bn, then any homeomorphism h : ?C ? ?D
extends to a homeomorphism h : C ? D.
Proof. We can assume that C = D = Bn. Then in vector notation, if x ? ?Bn, de?ne
h(tx) = th(x), 0 ? t ? 1.
De nition 2.5.5. Suppose B0,B1,...,Bn are tame 3-cells in S3 and for 0 < i ? n, the balls
Bi ? Int(B0) are disjoint. Then we call Cl(B0?
n?
i=1
Bi) an n-annulus with boundary ?Si,
where Si = ?Bi.
Lemma 2.5.6. Let A, A? be n-annuli in S3, with boundaries ?Si, ?S?i, and h : S0 ? S?0 be
a homeomorphism. Then h can be extended to a homeomorphism h : A ? A? with h(Si) = S?i.
5
Proof. By theorem 2.5.3, this is true for 1-annuli. Suppose it holds for (n-1)-annuli. Let
Bi, B?i be as in the previous de?nition. By theorem 2.5.3, there are homeomorphisms f :
Cl(B0 ? B1) ? S2 ? I and f? : Cl(B?0 ? B?1) ? S2 ? I such that f(S0) = S2 ? {1} and
f?(S?0) = S2 ? {1}. Let g = f? ? h ? f?1. Then g is a homeomorphism of S2 ? {1} onto
itself, and g can be extend to a homeomorphism g : S2 ?I ? S2 ?I. Let o be the origin of
R3; C be an in?nite cone with vertex o, whose intersection with S2 ?I is a 3-cell C disjoint
from S2,...,Sn, and g(C) is disjoint from S?2,...,S?n. Let D be the 3-cell Cl(S2 ?I ?C), D?
be g(D), and g1 be g restricted to ?D. Then by the induction hypothesis, g1 extends to a
homeomorphism g1 : Cl(D ?
n?
i=2
f(Bi)) ? Cl(D? ?
n?
i=2
f?(B?i)), mapping f(Si) to f?(S?i) for
1 < i ? n. De?ne eg : f(A) ? f?(A?) by letting eg be g on C, be g1 on Cl(D?
n?
i=2
f(Bi)). Let
h be f??1 ?eg ?f.
2.6 Connected Sum of Knots
Remark 2.6.1. From here on, all knots are assumed to be oriented and in ori-
ented S3.
De nition 2.6.2. Suppose K is a knot and C is a topological ball in S3. We say (C,C?K)
is an S ball pair if C is tame and (C,C?K) is topologically equivalent to the canonical
ball pair (B3,B1).
De nition 2.6.3 (connected sum of knots). Suppose Ki is a tame knot in S3, (Ci,Ci?Ki)
is an S ball pair, Di is Cl(S3 ? Ci), and Ai = Di?Ki for i = 1,2. By theorem 2.5.2,
Di are 3-cells. Let f : D2 ? C1 be an orientation-preserving homeomorphism such that
f(?A2) = ?A1 and the simple closed curve A1?f(A2) is oriented. The connected sum of
the knots K1 and K2, written K1?K2, is the simple closed curve A1?f(A2) in S3.
Remark 2.6.4. Such an f exists, for there is an orientation-preserving homeomorphism
?D2 ? ?C1 mapping ?A2 onto ?A1, and it can be extended to get an f by lemma 2.5.4,
6
and 2.1.4. And it is easy to see that f can be extended to an orientation-preserving ambient
homeomorphism, so the knot type of K2 is preserved.
Proposition 2.6.5. K1?K2 is an oriented knot in oriented S3.
Proposition 2.6.6. Connected sum is well-de?ned for equivalence classes of tame oriented
knots in oriented S3.
Proof. Here the notation will be as in de?nition 2.6.3. Let K?1 be a knot equivalent to K1.
De?ne C?1, D?1, A?1 and f? accordingly. We want to show that K1?K2 is equivalent to K?1?K2.
Clearly, f? ? f?1 : (C1,f(A2)) ? (C?1,f?(A2)) is an orientation-preserving homeomorphism
of pairs. Let g : (D1,A1) ? (D?1,A?1) be an orientation-preserving homeomorphism equal
to f? ?f?1 on ?D1 = ?C1. Then h equal to f? ?f?1 on C1, g on D1 is an desired ambient
homeomorphism.
Corollary 2.6.7. Let Ki be equivalence classes of tame knots, then
1. K1?K2 = K2?K1
2. (K1?K2)?K3 = K1?(K2?K3).
3. K1?K2 = K1, where K2 is the unknot.
2.7 Knot Factorization
De nition 2.7.1. A prime knot is a non-trivial tame knot which is not the connected sum
of two non-trivial tame knots. Knots that are not prime are said to be composite.
Theorem 2.7.2. There exist in?nitely many inequivalent prime knots.
Proof. See [3].
De nition 2.7.3 (decomposing sphere system for a tame knot). Let Sj, 1 ? j ? m, be
disjoint tame 2-spheres embedded in S3, bounding 2m balls Bi, 1 ? i ? 2m. If Bi contains
only the s balls Bl(1),...,Bl(s) as proper subsets, Ri = (Bi?
s?
q=1
Int(Bl(q)) is called the domain
7
Ri. The spheres Sj are said to be decomposing with respect to a tame knot K in S3 if the
following conditions are ful?lled:
1. Each sphere Sj meets K transversely in two points.
2. The knot Ki, which is the union of the arc Ai = K?Ri, oriented as K, and arcs on the
boundary of Ri, is prime. Ki is called a prime factor of K determined by Bi.
We call S = {Sj|1 ? j ? m} a decomposing sphere system with respect to K; if K
is prime we put S = ?.
Remark 2.7.4. Ki does not depend on the choice of the arcs on ?Ri.
De nition 2.7.5. Two decomposing sphere systems S = {(Sj,K)} and S? = {(Sl?,K)} are
called equivalent if they de?ne the same (unordered) factors Ki.
Theorem 2.7.6. Any non-trivial tame knot K can be decomposed into a ?nite number of
prime knots K = K1?K2?...?Kn. Furthermore, the decomposition is unique up to order. That
is, if K = K1?K2?...?Kn = K?1?K?2?...?K?m are two decompositions, then n = m and Ki = K?l(i)
for some permutation l of {1,2,...,n}.
Proof. W.l.o.g, we can assume that K is a polygonal knot in PL S3. The ?rst part of the the-
orem is an easy consequence of the additivity of the genus of PL knots.The uniqueness of de-
composition is proved by showing that any two decomposing sphere systems S = {(Sj,K)},
S? = {(Sl?,K)} are equivalent. For a detailed proof, please refer to [3].
8
Chapter 3
Generalizing Some Concepts
3.1 In nite Connected Sum of Knots
De nition 3.1.1. Let K be a knot in S3. Suppose B0,B1,...,Bn are tame balls whose
boundary meets K transversely in two points. For 0 < i ? n, the balls Bi ? Int(B0) are
disjoint. Let eK be the knot which is the union of the arcs K?Cl(B0 ?
n?
i=1
Bi), oriented as
K, and arcs on the boundary of Cl(B0 ?
n?
i=1
Bi), oriented in the way that eK is oriented.
Then eK is called the knot determined by Cl(B0 ?
n?
i=1
Bi) and K.
Recall that Cl(B0 ?
n?
i=1
Bi) is an n-annulus as de?ned in de?nition 2.5.5. We de?ne a
0-annulus to be a tame ball in S3. The word annulus may refer to a k-annulus for any
k ? 0.
The notations are consistent from 3.1.2 to 3.1.6.
De nition 3.1.2. Let K be a knot in S3. Then K is the connected sum of tame knots if
(1) there is a ?nite or countable sequence S = {Sj} of disjoint tame 2-spheres embedded in
S3, with each Sj meets K transversely in two points.
(2) if {Bi} is a countable sequence of tame 3-cells such that {?Bi} is a subsequence of {Sj}
and Bi+1 ? Int(Bi), then ?Bi is a point.
(3) let B be a ball bounded by some element of S, then there are at most a ?nite number of
balls whose boundaries are elements in S, say, C1,...,Cn, that are outermost in B, in the
sense that Ci ? Int(B) and there does not exist a ball D ? Int(B) bounded by some Sk ? S
such that Ci ? Int(D).
(4) the notation here is as in (3). Cl(B ?
n?
i=1
Ci) determines a non-trivial tame knot, which
9
is called a factor of K.
If {Sj} is a countable sequence, we say K is the in nite connected sum of tame
knots. S = {Sj} is called a decomposing sphere system for K. If in (3) the tame knot
is also prime, then we call S = {Sj} a prime decomposing sphere system for K. We
may also call a prime decomposing sphere system a decomposing sphere system.
Remark 3.1.3. Note that (2) implies that if B is a ball bounded by some element of S, and
C ? Int(B) is another ball bounded by some element of S, then there is a ball D ? Int(B)
whose boundary is an element of S such that C ? Int(D) and D is outermost in B. Thus
this excludes the case where there are in?nitely many balls whose boundaries are elements in
S contained in Int(B), but no one is outermost in B.
Remark 3.1.4. In the next two chapters, we will give examples of constructing in?nite
connected sum of knots with tame knots.
Proposition 3.1.5. If {Sj} is a countable, then S3 ? ?Sj is the union of a countable
collection of disjoint open annuli {Ai} and a closed set W of totally disconnected points,
such that the boundaries of Cl(Ai) are elements of S and W is the set of wild points of K.
Moreover, each closed annulus Cl(Ai) determines a non-trivial tame knot.
Proof. By conditions (1),(2),(3)(see remark 3.1.3), S3??Sj contains the union of a countable
collection of disjoint open annuli {Ai} such that the boundaries of Cl(Ai) are elements of
S. Let W = S3 ??Sj ??Ai. By (4), each closed annulus determines a non-trivial tame
knot. By (3) and (2), there is a countable sequence {Bn} of tame 3-cells such that {?Bn} is
a subsequence of {Sj}, Bn+1 ? Int(Bn), and ?Bn is a point p. Then p is not in any Sj, for
the spheres are disjoint, nor in any Ai. So W ?= ?.
If p ? W, then there is a countable sequence {Bn} of tame 3-cells such that {?Bn} is a
subsequence of {Sj}, p ? Bn+1 ? Int(Bn), and by (2), ?Bn = p. So ?(Bn?K) = p, and
hence p ? K. And p is a wild point of K, for there are in?nitely many knots converging to
10
p. Conversely, if p is a wild point of K, then p is not in any Sj or Ai by (2),(3),(4). Hence
p ? W. So W is the set of wild points of K.
W is closed in K, for the set of points at which K is locally tame is open. Since any
two distinct points in W are separated by two disjoint balls, W is totally disconnected.
The following lemma can be obtained from the proof of proposition 3.1.5.
Lemma 3.1.6. If {Bn} is a countable sequence of tame 3-cells such that {?Bn} is a subse-
quence of {Sj}, Bn+1 ? Int(Bn), then ?Bn is a wild point of K. If p is a wild point of K,
then there is a countable sequence {Bn} of tame 3-cells such that {?Bn} is a subsequence of
{Sj}, Bn+1 ? Int(Bn), and ?Bn = p.
3.2 Some Lemmas
Lemma 3.2.1. Let K be a tame knot in S3, ? ? K be an arc, and bK be a factor of K.
Then there is a tame 3-cell C such that K?C = ? and the knot determined by C is bK.
Proof. Since bK is a factor of K, there is a tame 3-cell D such that the knot determined
by D and K is bK. Let D?K = ?. Since K is tame, there is an orientation-preserving
homeomorphism h : (S3,K) ? (S3,K) such that h(?) = ?. Let C = h(D).
Lemma 3.2.2. Let the notation be de?ned as in 3.1.1, and assume that eK is a non-trivial
tame knot. Let bK be a factor of eK. Let B be Cl(S3 ? B0) or an element in {B1,...,Bn}.
Then there is a a tame 3-cell C such that B ? Int(C), ?C??Bi = ? for 0 ? i ? n, and
the knot determined by Cl(C ?B) and K is bK.
Proof. W.l.o.g, assume that B is Cl(S3 ?B0). Let ? be eK??B, and ? be a subarc of eK
such that ???Bi = ? for 1 ? i ? n and ? ? Int(?). By lemma 3.2.1,there is a tame 3-cell
D such that eK?D = ? and the knot determined by D and eK is bK. W.o.l.g, we can assume
that ?D and ?Bi are polyhedra in PL S3 and ?D is in general position with ?Bi, 0 ? i ? n.
Then replace ?D by a sphere S that is disjoint from each ?Bi, obtained from ?D by a ?nite
11
number of pushes (ambient homeomorphisms with compact support) 1. Let C be the closure
of the component of S that contains B.
Remark 3.2.3. Let the notation be as in lemma 3.2. Let M be the union of elements in
a subset of {Cl(S3 ? B0)}?{B1,...,Bn}. With an argument similar to that in proposition
3.2, we can show that there is a a tame 3-cell C such that M ? Int(C), ?C??Bi = ? for
0 ? i ? n, and the knot determined by Cl(C ?M) and K is bK.
1See [2] for de nitions and details.
12
Chapter 4
The Wilder Knots
4.1 Wilder Connected Sum
De nition 4.1.1 (Wilder connected sum of knots). Suppose K1 is a non-trivial tame knot
in S3, C1,C2,... is a sequence of tame 3-cells such that Ci+1 ? Int(Ci), ?Ci = p ? K1, and
(Ci,Ci?K1) is an S ball pair for each i. Let {Ki} be a sequence of non-trivial tame knots in
S3, and (Cij,Cij ?Ki) be an S ball pair, with Ci1?Ci2 = ? for i = 2,3,..., j = 1,2. De?ne
Mi = Cl(S3 ?Ci1?Ci2), Ni = Cl(Ci?1 ?Ci), for i ? 2, and M1 = N1 = Cl(S3 ?C1). By
theorem 2.5.3, Mi, Ni are homeomorphic to S2?I for i ? 2. Let ?ij be the two arcs Mi?Ki,
?ij be the two arcs Ni?K1, i ? 2, where ?i1 is the arc exiting ?Ci1 and entering ?Ci2, ?i1 is
the arc exiting ?Ci?1 and entering ?Ci. De?ne fi : Mi ? Ni to be an orientation-preserving
homeomorphism such that fi(?Ci1) = ?Ci?1, and for ?xed ij, fi maps the end points of ?ij to
the end points of ?ij, for i = 2,3,..., j = 1,2. Let ?1 = M1?K1. The Wilder connected
sum of {Ki}, written K1?K2?..., is the simple closed curve ?1 ??fi(?ij)?{p} in S3.
Remark 4.1.2. To see the fi de?ned above exist, let S1, S2 be the two components of the
boundary of A = S2 ? I and pij ? Si, qij ? Si, i = 1,2,j = 1,2 are eight points. W.l.o.g,
we can assume that pij should be mapped to qij. It is easy to see that there are orientation-
preserving (OP) and orientation-reversing (OR) homeomorphisms of S1 onto S1 taking p1j
to q1j. Such an OPH (ORH) can be extended to an OPH (ORH), say h, of A onto itself.
Then keep the points on S1 ?xed and slide h(p2j) on S2 till it is mapped to q2j. Also, each
fi can be extended to an OP ambient homeomorphism, and so the knot type of each Ki is
preserved.
Proposition 4.1.3. K1?K2?... is an oriented knot in oriented S3.
13
Proposition 4.1.4. Wilder connected sum is well-de?ned for equivalence classes of tame
oriented knots in oriented S3.
Proof. Suppose {K?i} is a sequence of tame knots such that K?i is equivalent to Ki for each
i. De?ne M?i, N?i, ??ij, ??ij, f?i, ??1 accordingly. Let h1 : (N1,?1) ? (N?1,??1) be an orientation-
preserving homeomorphism(OPH). Suppose we have got an OPHhn : (
n?
i=1
Ni,?1?
n?
i=1
2?
j=1
fi(?ij)) ?
(
n?
i=1
N?i,??1 ?
n?
i=1
2?
j=1
f?i(??ij)). Then f?n+1?1hnfn+1 : ?C(n+1)1 ? ?C?(n+1)1 is an OPH that maps
Kn+1 ??C(n+1)1 to K?n+1 ??C?(n+1)1, and it can be extended to an OPH g : Mn+1 ? M?n+1,
which maps Kn+1 ??C(n+1)2 to K?n+1 ??C?(n+1)2 as shown in the previous remark. Then g
can be modi?ed to get an OPH eg : (Mn+1,
2?
j=1
?(n+1)j) ? (M?n+1,
2?
j=1
??(n+1)j). De?ne hn+1
by letting hn+1 = hn ? f?n+1egfn+1?1. Finally, de?ne h to be hn on
n?
i=1
Ni for each n, and
h(p) = p?.
4.2 The Wilder Knots
De nition 4.2.1. A knot K in S3 is a Wilder knot if it is an in?nite connected sum of
tame knots with one wild point.
Remark 4.2.2. Intuitively, we can think a Wilder knot as a simple closed curve with a
sequence of tame knots tied in it convergent to a point p in it, possibly from each side of p.
See ?gure 4.1.
De nition 4.2.3. A wild knot is mildly wild if it is the union of two tame arcs.
Proposition 4.2.4. A Wilder knot is a mildly wild knot.
Remark 4.2.5. There is a mildly wild knot with one wild point that is not a Wilder knot.
See [8].
Proposition 4.2.6. Let {Ki} be a sequence of prime knots in S3. The Wilder connected
sum K1?K2?... is a Wilder knot. Conversely, every Wilder knot is the Wilder connected sum
of a sequence of prime knots.
14
Proof. The ?rst part is clear. Let K be an in?nite connected sum of tame knots with one
wild point p. Let S = {Sj} be a decomposing sphere system for K. By lemma 3.1.6, there
is exactly one countable sequence {Bn} of tame 3-cells such that {?Bn} is a subsequence
of {Sj}, Bn+1 ? Int(Bn), and ?Bn is a point, which should be p. By de?nition 3.1.2, the
knots determined by Cl(S3 ? B1) and Cl(Bn ? Bn+1) are tame. By theorem 2.7.6, every
tame knot can be factored into prime knots. Then the result follows from lemma 3.2.
Figure 4.1: Wilder knot
De nition 4.2.7. A prime knot type is an in nitely occurring prime if it appears in?nite
times in (a decomposition of) a wild knot, otherwise, a nitely occurring prime. We say
two wild knots have the same list of prime knots if and only if they have the same
set of ?nitely and in?nitely occurring primes, with each ?nitely occurring prime appearing
exactly the same number of times. We say two wild knots almost have the same list of
prime knots if and only if they di?er by a ?nite number of ?nitely occurring primes.
When working on the following theorem, we referred to the paper [5] by Fox and Harrold.
15
Theorem 4.2.8. Two Wilder knots K and K? are equivalent if and only if they have the
same list of prime knots.
Proof. Suppose K and K? are equivalent. By proposition 4.2.6, K can be factored into prime
knots. We show that K is uniquely factored into prime knots, and it would follow that K
and K? have exactly the same list of prime knots. Assume that K has n type-? knots in
a decomposition. Take a tame 2-sphere S that separates the wild point p of K from the n
type-? knots of K, and intersects K transversely in two points. Denote the closure of the
complementary component of S that does not contain p as D1, and the closure of the other
complementary component of S as D2. Let K1 be the knot determined by D1. Suppose in
a di?erent decomposition, K has m type-? knots. Take a tame 2-sphere S? ? Int(D2) that
separates the wild point p of K from the m type-? knots of K and intersects K transversely
in two points. Let D?1 be the closure of the complementary component of S? that does not
contain p, and K?1 be the knot determined by D?1. Then K1 is a factor of K?1, and hence
m can not be less than n, for otherwise K?1 would not be uniquely factored. By symmetry,
m = n. If K has in?nitely many type-? knots in one decomposition, then for each n, K has
no less than n type-? knots in a di?erent decomposition, so K has in?nitely many type-?
knots in the other decomposition.
Conversely, suppose K and K? have the same list of prime knots. By proposition 4.2.6,
K = K1?K2?... and K? = K?1?K?2?..., where {Ki} and {K?i} are two sequences of prime
knots. Since Wilder knots are uniquely decomposed into prime knots, K = K1?K2?... and
K? = K?1?K?2?... have the same list of prime knots. If we can show that there is a decomposing
sphere system for K such that K = K?1?K?2?..., then K and K? are equivalent by proposition
4.1.4. Let p be the wild point of K. Fix an n, there is a tame 2-sphere S1 that meets K
transversely in two points and separates K?1,K?2,...,K?n from p. Using lemma 3.2 repeatedly,
there are tame 3-cells C1,C2,...,Cn with Ci+1 ? Int(Ci), S1 ? Int(Cn), and the knot
determined by S3?Int(C1) is K?1, the knot determined by Ci?Int(Ci+1) is K?i+1, 1 ? i < n.
Next we take a tame 2-sphere S2 that meets K transversely in two points and separates
16
K?n+1,K?n+2,...,K?2n from p. If S1 separates K?n+1,K?n+2,...,K?2n from p, we take S2 = S1. By
lemma 3.2, there are tame 3-cells Cn+1,Cn+2,...,C2n with Ci+1 ? Int(Ci), S2 ? Int(C2n),
and the knot determined by Ci ?Int(Ci+1) is K?i+1 for n ? i < 2n. Continuing in this way,
we can ?nd a sequence of tame 3-cells C1,C2,... such that Ci+1 ? Int(Ci), and ?Ci = p,
and the knot determined by S3 ?Int(C1) is K?1, the knot determined by Ci ?Int(Ci+1) is
K?i+1.
17
Chapter 5
C-wild Knots
5.1 Cantor Connected Sum
De nition 5.1.1 (Cantor connected sum of knots). Suppose K01 = K0 is a non-trivial tame
knot in S3, Cij, i = 1,2,..., j = 1,2,...,2i is a sequence of tame 3-cells such that C1j are
disjoint in S3 and C(i+1)(2j?1), C(i+1)2j are disjoint subsets of Int(Cij), (Cij,Cij ?K1) is an
S ball pair for each ij, and diamCij ? 0 as i ? ?. Let {Kij} be a sequence of non-trivial
tame knots in S3, and (Cijk,Cijk ?Kij) be disjoint S ball pairs for ?xed ij, i = 1,2,...,j =
1,2,...,2i,k = 0,1,2. De?ne Mij = Cl(S3 ?
2?
k=0
Cijk), Nij = Cl(Cij ?C(i+1)(2j?1)?C(i+1)2j),
for i ? 1, and M0 = N0 = Cl(S3?C11?C12). By lemma 2.5.6, Mij, Nij are homeomorphic
for i ? 1. Let ?ijk be the three arcs Mij ?Kij, ?ijk be the three arcs Nij ?K0, where ?ij0 is
the arc exiting ?Cij0, ?ij2 is the arc entering ?Cij0, ?ij0 is the arc exiting ?Cij, ?ij2 is the arc
entering ?Cij. De?ne fij : Mij ? Nij to be an orientation-preserving homeomorphism such
that fij(?Cij0) = ?Cij, and for ?xed ijk, fij maps the end points of ?ijk to the end points
of ?ijk, for i = 1,2,..., j = 1,2,...,2i, k = 0,1,2. Let ?0 = M0?K0. Let Ci = ?
j
Cij. The
Cantor connected sum of {Kij}, is the simple closed curve ?0 ??fij(?ijk)??Ci in S3.
Remark 5.1.2. With lemma 2.5.6 and an argument similar to that in remark 4.1.2, we can
show that fij de?ned above exist and can be extended to an OP ambient homeomorphism.
Proposition 5.1.3. The Cantor connected sum is an oriented knot in oriented S3.
Proposition 5.1.4. Cantor connected sum is well-de?ned for equivalence classes of tame
oriented knots in oriented S3.
Proof. Let Mi = ?
j
Mij, Ni = ?
j
Nij, fi = ?
j
fij. Then use an argument analogous to the
proof of 4.1.4.
18
Figure 5.1: Cantor connected sum
5.2 C-wild knots
De nition 5.2.1. A Cantor set is a compact metrizable space that is totally disconnected
and has no isolated points.
De nition 5.2.2. A C-wild knot is an in?nite connected sum of tame knots whose wild
points form a Cantor set.
Proposition 5.2.3. Let K be a C-wild knot and W be the set of wild points of K. Then
K ?W is a countable union of open tame arcs {Ai} such that Cl(Ai) is a tame arc whose
end points are wild points of K.
19
Proposition 5.2.4. A C-wild knot is the Cantor connected sum of a sequence {Kij} of
prime knots, where i = 0,1,..., j = 1,2,...,2i. Conversely, the Cantor connected sum of a
sequence of prime knots is a C-wild knot.
Proof. The second part is clear. The ?rst part is clear by proposition 5.2.3, theorem 2.7.6
and lemma 3.2. There is a detailed proof in theorem 5.3.7.
Proposition 5.2.5. A C-wild knot can be and is uniquely decomposed into prime knots.
Proof. The existence part is by proposition 5.2.4. With an argument similar to that in the
proof of proposition 4.2.8, we can show that every C-wild knot is uniquely decomposed into
prime knots.
De nition 5.2.6. Let K be a C-wild knot. Let Cij, i = 1,2,..., j = 1,2,...,2i be a sequence
of tame 3-cells such that C1j are disjoint in S3 and C(i+1)(2j?1), C(i+1)2j are disjoint subsets
of Int(Cij). We say C = {?Cij} is a Cantor decomposing sphere system (Cdds) for
K if C is a decomposing sphere system for K.
Theorem 5.2.7. Two C-wild knots K and K? are equivalent if and only if there are prime
Cantor decomposing sphere systems {?Cij} and {?C?ij} for K resp. K? such that
1. K and K? have the same list of prime knots.
2. ?x ij, the C-wild knots Jij and J?ij determined by Cij and C?ij respectively almost have the
same list of prime knots.
Proof. Suppose K and K? are equivalent. W.l.o.g, we can assume that K = K? as sets.
Let W be the set of wild points of K. Let {?Cij} and {?C?ij} be two Cdds for K such
that Cij ?W = C?ij ?W. By proposition 5.2.5, K is uniquely decomposed into prime knots.
And it is easy to see that for ?x ij, the C-wild knots Jij and J?ij determined by Cij and C?ij
respectively almost have the same list of prime knots.
Conversely, suppose there are Cdds {?Cij} and {?C?ij} for K resp. K? satisfying con-
ditions 1,2, and K and K? are the Cantor connected sum of {Kij} resp. {K?ij}. We want
20
to ?nd a Cdds ?Dij for K such that K is the Cantor connected sum of {K?ij}, then K
would be equivalent to K? by proposition 5.1.4. By 1, 2, for a large enough n, K?01 and
the prime knots appearing in J?1j but not in J1j, for j = 1,2, would appear in the tame
knot determined by Cl(S3 ?
2n?
j=1
Cnj). By remark 3.2.3, we can ?nd D1j such that the knot
determined by D1j is J?1j and the knot determined by Cl(S3 ? ?D1j) is K?01, j = 1,2.
Next we want to ?nd D21 and D22 such that the knot determined by D2j is J?2j and the
knot determined by Cl(D11 ? (D21 ? D22)) is K?11, j = 1,2 . If all the prime knots ap-
pearing in Jn1 ? Jn2 ? ... ? Jn2n 2 also appear in J?21, and all the prime knots appearing in
Jn(2n 2+1) ?...?Jn2n 1 also appear in J?22 then we can ?nd D21 and D22 using remark 3.2.3.
If not, Since J2j and J?2j, almost have the same list of prime knots, for m large enough, all
the prime knots appearing in Jm1 ?Jm2 ?...?Jm2m 2 also appear in J?21, and all the prime
knots appearing in Jm(2m 2+1) ?...?Jm2m 1 also appear in J?22. Then we can ?nd D21 and
D22 by applying remark 3.2.3. Continuing this way, we can ?nd a desired Cdds {?Dij} for
K.
5.3 Wilder Knots in C-wild Knots
De nition 5.3.1. Let K be a C-wild knot and p be a wild point of K. Let S = {Sk} be a
decomposing sphere system for K. A Wilder knot is called a Wilder knot (relative to
S) in K with wild point p if it is the union of subarcs of K and arcs on a subsequence
of {Sk} and its wild point is p.
Proposition 5.3.2. The notations are de?ned as in de?nition 5.3.1. Then there is a Wilder
knot in K with wild point p.
Proof. By lemma 3.1.6, there is a countable sequence {Bn} of tame 3-cells such that {?Bn}
is a subsequence of S = {Sk}, Bn+1 ? Int(Bn), and ?Bn = p. We can assume that Bn+1 is
outermost in Bn. Let Mn+1 be the union of all the balls that are outermost in Bn, and whose
boundaries are elements of S. Let Kn be the tame knot determined by Cl(Bn?Mn+1). Then
the in?nite connected sum of {Kn} is a desired Wilder knot.
21
Remark 5.3.3. Note that there are in?nitely many inequivalent Wilder knots in K with
wild point p.
Theorem 5.3.4. Two C-wild knots K and K? are equivalent if and only if there are prime
Cantor decomposing sphere systems {?Cij} and {?C?ij} for K resp. K? such that
1. K and K? have the same list of prime knots.
2. let Jij and J?ij be the C-wild knots determined by Cij and C?ij respectively. Fix ij and let ?
be a Wilder knot in Jij. Then there is a Wilder knot ? in J?ij such that all but ?nitely many
prime factors of ? are prime factors of ?.
Proof. Suppose K and K? are equivalent. W.l.o.g, we can assume that K = K? as sets. Let
W be the set of wild points of K. Let {?Cij} and {?C?ij} be two Cdds for K such that
Cij ?W = C?ij ?W. Let ? be a Wilder knot in Ji0j0 with wild point p. By lemma 3.1.6, there
is a countable sequence {Bn} of tame 3-cells such that {?Bn} is a subsequence of {?Cij},
Bn+1 is outermost in Bn, and ?Bn = p. Suppose ?Bn ? ? ?= ?. Let ?n be the tame knot
determined by Cl(Bn ?Bn+1) and ?. Assume that ? is the Wilder connected sum of {?n}.
Further assume that Ci0j0 = C11 and Bn = Cn1 as sets. For each n ? 2, take mn large enough
so that ?n?1 is a factor of the tame knot determined by Cl(C(n?1)1 ?Cn1 ?
2mn n+1?
j=2mn n+1
C?mnj)
and K. Let ? be the knot that is the union of subarcs of J?11 and arcs on the boundaries of
{C?mnj}.
Conversely, suppose there are prime Cdds {?Cij} and {?C?ij} for K resp. K? such that
conditions 1, 2 are satis?ed. Suppose that there are in?nitely many prime factors of Ji0j0 that
are not prime factors of J?i0j0. Then we can ?nd a Wilder knot ? in Ji0j0 such that in?nitely
many prime factors of ? are not prime factors of J?i0j0. But this contradicts condition 2. So
condition 2 of proposition 5.2.7 holds. So K and K? are equivalent.
De nition 5.3.5. A cyclic order on a set X is a relation, written [a,b,c], that satis?es
the following axioms:
Cyclicity: If [a,b,c] then [b,c,a]
22
Asymmetry: If [a,b,c] then not [c,b,a]
Transitivity: If [a,b,c] and [a,c,d] then [a,b,d]
Totality: If a, b, and c are distinct, then either [a,b,c] or [c,b,a]
De nition 5.3.6. A function between two cyclically ordered sets f : X ? Y is called
order-preserving if [a,b,c] implies [f(a),f(b),f(c)].
Theorem 5.3.7. Two C-wild knots K and K? with prime decomposing sphere systems S =
{Sk} resp. S? = {S?k} are equivalent if and only if
1. K and K? have the same list of prime knots.
2. let W, W? be the set of wild points of K resp. K?. Then there is an order-preserving
bijection f : W ? W? such that if p, q are the end points of a tame subarc of K, then f(p),
f(q) are the end points of a tame subarc of K?.
3. suppose ? is a Wilder knot in K with wild point p. Then there is a Wilder knot ? with
wild point f(p) in K? such that all but ?nitely many prime factors of ? are prime factors of
?.
Proof. Assume conditions 1, 2, 3. If x ? W, then x? denote f(x) ? W?. Let V be the set of
wild points of K that are end points of tame subarcs of K.
Step 1. ?nd a prime Cdds for K.
Let pi,qi ? W be the end points of a tame subarc ?i of K for i = 1,2 such that [p1,p2,q2]
and [p1,q2,q1]. Let C11 be a tame 3-cell that intersects K transversely in two points on
Int(?1) resp. Int(?2) such that p1,p2 ? Int(C11), and C12 be a tame 3-cell that intersects K
transversely in two points on Int(?1) resp. Int(?2) such that q1,q2 ? Int(C12). Moreover,
C11 and C12 are disjoint, and the knot determined by Cl(S3?(C11?C12)) is prime, which is
possible by lemma 3.2. Let ?3 ? Int(C11) be a tame subarc of K with end points p3,q3 ? V
such that [p1,p3,q3] and [p1,q3,p2]. Let C21 ? Int(C11) be a tame 3-cell that intersects
K transversely in two points on Int(?1) resp. Int(?3) such that p1,p3 ? Int(C21), and
C22 ? Int(C11) be a tame 3-cell that intersects K transversely in two points on Int(?3)
23
resp. Int(?2) such that q3,p2 ? Int(C22). Moreover, C21 and C22 are disjoint, and the knot
determined by Cl(C11 ?(C21 ?C22)) is prime. Continuing in this way, we can ?nd a prime
Cdds {?Cij} for K.
Step 2. ?nd a corresponding prime Cdds for K?.
By condition (2), p?i,q?i ? W? are the end points of a tame subarc ??i of K? for i = 1,2 such
that [p?1,p?2,q?2] and [p?1,q?2,q?1]. Let C?11 be a tame 3-cell that intersects K? transversely in two
points on Int(??1) resp. Int(??2) such that p?1,p?2 ? Int(C?11)... Continuing in this way, we
can ?nd a prime Cdds {?C?ij} for K?.
Step 3. show that prime Cdds C = {?Cij} for K, C? = {?C?ij} for K? satisfy conditions 1, 2
of proposition 5.3.4.
Condition 1 of 5.3.4 is true by proposition 5.2.5, and condition 1 of this theorem. Now
we prove condition 2 of 5.3.4. Let Jij and J?ij be the C-wild knots determined by Cij and
C?ij respectively. Let ? be a Wilder knot in Ji0j0 with wild point p. By lemma 3.1.6,
there is a countable sequence {Cm} of tame 3-cells such that {?Cm} is a subsequence of
{?Cij}, Cm+1 ? Int(Cm), and ?Cm = p. We can assume that Cm+1 is outermost in Cm,
?Cm?? ?= ?, and ?C1 is also an element of S. Let ?m be the tame knot determined by
Cl(Cm ? Cm+1) and ?. Then we can ?nd Bm1,...,Bmnm whose boundaries are elements of
S such that ?m is a factor of the tame knot determined by Cl(Cm ? Cm+1 ?
nm?
t=1
Bmt) and
K. Let ? be the knot that is the union of subarcs of K and arcs on the boundaries of C1
and {Bmt}, where 1 ? t ? nm, m ? 1. Then ? is a Wilder knot relative to S in K with
wild point p such that all but a ?nite number of prime factors of ? are prime factors of ?.
By condition 3, there is a Wilder knot ? relative to S? with wild point p? in K? such that
all but ?nitely many prime factors of ? are prime factors of ?. Then we can ?nd a Wilder
knot ? relative to C? with wild point p? in K? such that all but ?nitely many prime factors
of ? are prime factors of ?. By step 2, p? is in C?i0j0. So we can assume that ? is in J?i0j0. So
conditions 1, 2 of 5.3.4 are satis?ed and hence K and K? are equivalent.
24
Conversely, assume that K and K? with prime dds S = {Sk} resp. S? = {S?k} are
equivalent. Condition 1 is by proposition 5.2.5. Condition 2 follows immediately. Condition
3 can be done similarly as in Step 3.
25
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