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Journal of Knot Theory and Its Ramifications, Vol. 2 , No. 3 (1993) 251-284
@World Scientific Publishing Company
REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND
T H E I R INTERPRETATION AS MOVES T O MOVIES
[
J. SCOTT CARTER
Department of Mathematics, University of South Alabama,
Mobile, Alabama 36688, [email protected]
1
MASAHICO SAITO
Department of Mathematics, University of Texas,
A u s t i n , Tezas 78712, saitoQmath.utezas.edu
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Received 6 March 1993
ABSTRACT
A movie description of a surface embedded in Cspace is a sequence of knot and link
diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional
cross sections perpendicular to a fixed direction. In the cross sections, an immersed
collection of curves appears, and these are lifted to knot diagrams by using the projection
direction from 4-space. We give a set of 15 moves to movies (called movie moves) such
that two movies represent isotopic surfaces if and only if there is a sequence of moves
from this set that takes one to the other. This result generalizes the Roseman moves
which are moves on projections where a height function has not been specified.
The first 7 of the movie moves are height function parametrized versions of those
given by Roseman. The remaining 8 are moves in which the topology of the projection
remains unchanged.
1. Introduction
Knotted surfaces can be studied diagrammatically by their projections in 3space or by their movies. Having chosen a projection from 4-space into bspace,
the projected surface can be cut by 2-dimensional planes that are perpendicular
to a fixed direction. The general intersection of such a plane with the projected
surface is a collection of immersed curves. Over and under crossing information is
depicted on these curves by breaking the resulting diagrams at their double points.
Here, over and under refers the relative height in 4-space as measured with respect
to the projection from 4 to 3-space where 3-space is regarded as a subspace of 4space. A movie is this sequence of classical link diagrams. The difference between
two successive stills in a movie is either a topological equivalence of the underlying
graph or an elementary string interaction: A birth or death, a Reidemeister move
of type I, 11, or 111, or a surgery. Between the stills that so differ, a critical point
of the surface or a critical point of the double points of the surface occurs. The
critical data are depicted via their effect on the cross sectional diagrams.
252
J. S. C a r t e r €4 M. Saito
From such a movie, a projection can be constructed. And a projection together
with crossing information yields a movie. That the movie can be slowed to be
realized as sequence of elementary string interactions is one consequence of the
work herein.
From the diagrammatic point of view one is lead to ask: "What are the Reidemeister moves for movies?" (This question was posed to the first named author by
Louis Kauffman during the "New Mathematical approaches to DNA" conference
held in Santa Fe, New Mexico in January 1990. We are grateful for the question
and his encouragement .)
Movie diagrams can be considered as projections of knotted surfaces with an
explicit height function on the 3-space and so Reidemeister moves on movies must
include the ability to change height functions. In t h e classical c a s e (knots and links
of circles in 3-space) this problem is explicitly solved by including the A s (or simply
A-type) moves [lo] [6]. These are moves in which the topology of the 4-valent graph
underlying the diagram remains unchanged.
Here we give a proof of the 4-dimensional Reidemeister Theorem due to Roseman
[ll, 12, 131. (See Homma-Nagase [8] for the case of PL maps in 3-manifolds.)
Moreover, we give a list of movie moves that are sufficient to express any isotopy.
In the process, we prove the sufficiency of the type I, type 11, type 111, and A-type
moves to express classical knot isotopies in the presence of a height function. (See
Figures 10 through 14.) That these techniques could be applied to the classical case
was first noticed by Roseman [ll].
One might expect that the known solutions [4] [14] to the Zamolodchikov equation yield invariants of knotted surfaces analogous to the Jones polynomial. This
is because the Zamolodchikov equation is essentially the quadruple point move of
Figures 29 and 22. For these solutions to define invariants of knotted surfaces, one
must have a set of moves in hand to check invariance under isotopy. In fact, some
of the movie moves are problematic if we use Zamolodchikov's assignment of amplitudes to the elementary string interactions. See [2] for further discussion, and see
[3] for an interpretation of the movie moves in the language of 2-categories.
1.1. Statements of results and summary
Section 2 contains the background on general position maps that will be assumed
throughout. The notation needed to distinguish various multiple points and their
covers is introduced. The main result of the section is the commutative diagram 2.1
that is an immediate consequence of the definitions. This material is folklore that
should be in the literature.
In Sec. 3, we prove that a given general position map (of curves, surfaces, or
solids) or a general position immersion in higher dimensions possesses a function
that is Morse on all of its multiple points. In Sec. 4, we show how to project t o a
plane and obtain fold lines that iniersect the self intersection manifolds transversely.
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253
The results of these two sections are the key technical steps in finding a sufficient
set of movie moves.
Section 5 contains the moves on knots that we call elementary string interactions. These moves are analogous to crossings and optima of classical knots.
General position surfaces in 3-manifolds can be decomposed as elementary string
interactions. These decomposition theorems appear in Theorem 5.2 and the
remarks 5.2.1. Theorem 5.1 is the classical Reidemeister Theorem.
In Sec. 6 the Roseman moves of isotopy are listed. We use the results of
Secs. 3 and 4 to assume the isotopy between knottings is in an appropriate
position. Then it is decomposed as a sequence of the Roseman moves. The proof
follcws from studying the involution on the double decker set of a projection of the
isotopy.
In Sec. 7, Roseman's moves and some others are interpreted in terms of movies.
Finally, these 15 moves are sufficient to construct any isotopy of movies of a knotted
surface (Theorem 7.1). This, the main point of the paper, was announced in [2].
2. The M u l t i p l e P o i n t S e t s
A unified treatment to the various multiple point sets is given here. To define
these we can assume all the maps are nice in the following sense. First, all maps
are smooth throughout the paper. Next, it is known that any smooth map can
be approximated (by a homotopy) by a map with simple singularities. Specifically,
we can assume that a map O : Xn -+ Yntl has, at worst, branch points: In case
n = 2, in a neighborhood of a branch point the map looks like Whitney's umbrella
which is parametrized by (x, y) I+ (x, xy, y2). In case n = 3, in a neighborhood of
the branch point the map looks like this Whitney's umbrella crossed with an open
interval. See [7] for details. Furthermore, any immersion can be approximated by
a nice one which locally looks like the intersection of coordinate planes.
a general position map if for any point in the image of O there
Thus we call
is a neighborhood in which the image of O is either
(1) the intersection of the coordinate planes (including the embedded neighborhood), or
(2) a standard (as above) branch point.
In case n = 3 we further assume that the branch point set (which is 1 dimensional) intersects the image of O transversely.
We give notations for multiple point sets and their preimages for general position
maps. For the rest of the section all maps are general position maps.
Define T@(r)to be the closure of the configuration space
-
-
C@(r)= { ( x l , . . . , x r ) E X x . . . x X : xi
-
# x,
for i
# j, & O(xl) = . . . @(x,))
.
r
+
The set T@(r)is a manifold of dimension n 1 - r . Since O may have branch
points, it is convenient to include these limit points; in this way if X is compact
then T@(r)is compact while C@(r)may not be.
-
J . S . Carter tY M. Saito
254
Evidently, C@(r)admits a free action of the symmetric group C, that extends
I
V
to an action on T+(r). The fixed points of this action project to branch points of
the map @. Define a space
-
This is a quotient of the product space modulo the given action of C, on T+(r) and
the permutation action on the set of all k-fold subsets of {1,2,. . . , r}. Specifically,
an equivalence relation is defined by:
(x,,, . . . , xnV)x {ii, . . . , ik}
N
>
I
( x m l , .. . , xm,) x {jl,. . . , j k }
if and only if there is a permutation a E C, such that
~ ( { i l ,.. . , ik}) = {jl,. . . ,j k )
and
(xm17 . . . , xmP)= (xo-l(nl)>.. . 3 xo-1(nv)) .
For 0 5 k
defined by
< r there is a (branched) covering
-
I
T+(r, r ) = T+(r)
~ , , k : T+(r, k) -+
rr,k: [(xi,. .. , x r ) ; {jl,... , j k l l
[ X I , . . ., x r ]
where T@(r)has been identified with T@(r)/C,. The cardinality of the general fibre
is (L).
: T@(r,k) -+ T+(k) that is defined by
There is also an immersion
Any of the sets T@(r)map into Y via @, : [(XI,.. . , x,); {jl,. . . ,jk}] H @(XI).We
adopt the conventions that T+(O) = Y , T+(1) = X , and cPo is the identity map on
Y. In the next section, we let X, denote the image @,(T+(r)).
That these maps are well defined follows directly from the definitions except
possibly at branch points. But by dimension restrictions, the branch points occur
only for r = 2, and so they lie on the diagonal of X x X.
2.1. Proposition. For any r = 1 , 2 , . . . , ( n
commutative diagram:
+ 1 ) and for
Proof. The proof follows directly from the definitions.
any 0
< k 5 r , there is a
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255
The set T@(r,1) = T@(r,r - 1) is called the r-decker set (or manifold) and the
set T+(r) is called the r-tuple set (or manifold). They are immersed manifolds from
the assumption that cP is a general position map.
Consider for example the intersection of the four coordinate 3-balls in the 4ball. This is the image of @ : B: U B; U Bi U Bi -t B4.(Here Bf = {(x2,x3, x4) :
C xj" 5 11, Bz = {(XI,xg, 24) : C x; 5 11, and so forth; the map P h i includes B?
into B4 by setting the ith coordinate equal to 0.) The quadruple point manifold,
T@(4),is a single point; its quadruple decker manifold, Ta(4, I), is the disjoint union
of four points. The triple point manifold, T@(3)consists of four coordinate arcs; the
triple decker manifold Ta(3, 1) then has twelve components - three over each component of Ta(3). The six coordinate disks form the double point manifold, Ta(2),
and the double decker manifold Ta(2, 1) contains two copies of each of these. The
six points in Ta(4,2) immerse in T@(2)as the set of double points of @3,2(Ta(3,2)).
In Fig. 1 coordinate squares in the unit cube in 3-space are depicted. The triple
point Ta(3) is denoted by Ut1,2,3). The double point set Ta(2) consists of three
lines indicated by Utl,2) (resp. U{2,31,U{1,3)) each of which is the intersection line
between the squares Utl1 and Ut2) (resp. Ut2) and Ut3), UilI and Ut3)).
Fig. 1. Coordinate neighborhoods of multiple points
3
2
Remark. In studying the bordism group of immersions, one can include normal
bundle information with these covers and obtain a geometric realization of transfer
maps in stable homotopy.
!
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J. S. Carter €4 M. Saito
3. E v e r y w h e r e M o r s e Functions
In this section X and Y are compact manifolds. Let O : ( X , doX, d l X ) -+
(Y, doY, 8lY) denote a proper general position map if n = 1, 2, 3 or a general
position immersion if n 4. Both the boundary of X (resp. Y) are written as the
union of two disjoint sets each of which is the union of components of the boundary
of X (resp. Y): d X = doX U dlX, (resp. dY = doY U dlY).
A map f : (Y, doY,dlY) + (I,(01, {I)), from the space Y to the unit interval
I = [O, 11, is a proper everywhere Morse function if
>
1. f oO, is Morse on T*(r) for all r = 0 , 1 , ... , n + l ) ,
2. (f o 0,)-'(i) = diX, for i = 0,1, and
3. (f o a,) has no critical points in a neighborhood of d X r , for all r = 0 , 1 , . . . ,
n f 1.
If X and Y are closed manifolds, then
is called an everywhere Morse function
if O satisfies the first condition in the above. By definition, the 0-tuple manifold,
T@(O)is Y, and a. is the identity map.
3.1. T h e o r e m . Let O : ( X ; doX,d l X ) + (Y; doY, dlY) be a proper general position
map from a compact n-manifold, X, to a compact ( n 1)-manifold Y. Then the
set of proper everywhere Morse function f : (Y; doY, dlY) + ( I ; {O), (1)) is dense
in the C 2 topology.
+
3.1.1. R e m a r k . By the commutativity of Diagram 2.1, an everywhere Morse
function is equivariantly Morse on each of the T@(r,k).
Theorem 3.1 holds when the boundaries of X or Y are empty.
3.1.2. C o o r d i n a t e Neighborhoods. Before proving 3.1, we establish some notation with regard t o coordinate neighborhoods. Let U denote a coordinate neighborhood of a point y € Y , and let h : U --+ Rn+l,denote a coordinate function such
that h(y) = 0. If y E dY, then without loss of generality we may assume that h
maps U to the intersection of the interior of the unit ball and the upper-half space:
{(XI,.. . , x n + l : xn+l 0).
If y is a branch point of O, then there is a unique x E X such that (1) O(x) = y,
and (2) there is a coordinate neighborhood (W, k) of x such that hak-'(xl, xz, (x3))
= (xl, ~ 1 x 2x,i , (x3)) when n = 2 (respectively n = 3).
If y is not a branch point, the coordinate neighborhood, U, can be chosen so
that O-l(U) is a collection of disjoint coordinate neighborhoods of points in X.
Thus
O-'(u) = u ~ U . . U U , , for 0 5 r 5 n + 1 ,
>
and if y E d Y then 0
5 r 5 n. Furthermore, there are coordinate maps
Finally, with U a s above, each of the multiple point manifolds has similar coordinate neighborhoods. Specifically, if S c {1,2,. . . , r ) is any subset, define
Movie Moves
257
Us = { Z E Rn+l: zj = 0 for j E S). Then Us is a coordinate neighborhood
of a point in T @ ( J S Jthat
)
maps into U . We adopt the conventions that UB = U .
For more intuition about these coordinatizations see Fig.1.
Proof of 3.1. In the Lemmas 3.1.4 through 3.1.10 we show that in the local
coordinates, the set of functions that are Morse on each of the self intersection sets
is open and dense in the set of all functions. This is relatively clear from the local
description of general position multiple or branch points, for almost all projection
directions are Morse on all the self intersection strata. From these local gener:;
position coordinates we can construct a properly everywhere Morse function as
follows. (The technical details of the proof are similar to those given in Milnor [9]
Theorems 2.7 and 2.5.)
First, the set of properly everywhere Morse functions is open in the C2 topology
by Lemmas 3.1.7 and 3.1.8.
Lemma 3.1.3 gives a function without any critical points on the boundaries of
T @ ( r )for r = 0,. . . , n + 1. We adjust this function in stages. First cover Y by a
finite number of coordinate neighborhoods in which @ has a standard form. Choose
a compact refinement of this cover. Then find a function fl that is everywhere
Morse on a given compact neighborhood, C1, and that is C2 close to the original
function. This can be done because the set of everywhere Morse functions is dense
for the local coordinates.
Any function that is sufficiently close to fl is also everywhere Morse on C l . Furthermore, there is a function f2 that is close to fl and that is everywhere Morse on
a compact coordinate neighborhood C 2 . The process continues for a finite number
of stages until we find a neighborhood of the given function such that any function g therein is everywhere Morse on all of Y. That is, g o Or is Morse for all
r = 0 , . . . , n 1. Thus the proof follows from the lemmas below.
+
When the proof of one of the following lemmas is an easy translation of the
proof given in Milnor [9], the details are omitted.
I
3.1.3. Lemma. There is a smooth function f : Y
I
Proof. See Milnor [9] Lemma 2.6.
-
[O, 11 such that
1. ( f o ~ , ) - ' ( i ) = d i T a ( r ) for r = 0 , 1 , . . . n, and i = 0 , l ;
2. There are no critical points o f f , f a , and f a , in a neighborhood of each of
the boundaries.
3.1.4. Lemma. Let U denote an open ball around 0 in Rn+'. Let f : U -+ R
denote any C2 map. T h e n almost all linear maps L : Rn+l --+ R, satisfy the
following condition: the composition ( f L ) Q s is Morse for all S C {1,2,. . . , r )
where r n 1 and as : Us
U is the inclusion.
< +
-
+
Proof. The proof is similar the proof of Lemma A in Milnor [9], but details are
provided.
258
J. S. Carter
€4 M. Saito
+
Let y E U be in the image of some Os. If IS1 = n 1, then by definition, f is
Morse.
Consider the manifold N = Us x H o r n R ( ~ " ,R ) where St = {1,2, . . . , n 1)\S,
and R" denotes the linear subspace of Rn+' in which xi = 0 if i E S . The manifold
N is of dimension 2(n 1- ISI).
Let M s = {(x, L) E N : D(f
L)as = 0); this is a submanifold that is
homeomorphic to Us by the map x I+ (x,-D(fOs)(x)). Each point (x,L) E Ms
corresponds to a critical point of (f L)Os that is degenerate when the matrix
+
+
+
+
is singular.
The projection T : M + H o m ( ~ " , R ) onto the second factor is the map x H
-D(fQs). Thus (f L)@s has a degenerate critical point if and only if LIRSl is
the image of a critical point of T. Now by Sard's theorem the set of maps, L', that
are the image of critical points of T : M + H O ~ ( R ~R ') ,has measure zero in R".
Thus if (f + L)Qs has a degenerate critical point, then LIRsr is in this measure
zero subset of R ~ ' .So for fixed S, the set of Ls (ells) that yield degeneracy is of
measure zero in R n t l . The union over all S C {1,2,. . . , r } of these measure zero
sets is still of measure zero, and this completes the proof.
+
3.1.5. Lemma. Let O : R2 + R3 be defined by @(x,y ) = (x,xy, y2). Let f :
R 3 + R be a C m function. T h e n for almost all linear maps L : R3 + R , the map
(f L)O has only non-degenerate critical points.
+
Proof. The only critical point of O is (0,O). By Lemma 3.1.4, we may assume that
(f
L) is Morse, and so (f L) o O is Morse except possibly a t (0,O).
The map f can be C 2 approximated by a second degree polynomial function
f = 3 air; +
aijxizj. Let L =
biz; denote a linear map. Then
L) has a critical point at (0,O) if and only if bl = -al. It is degenerate if and
(f
only if b3 = -a3. SO the set of linear maps that give degenerate critical points is of
measure zero.
+
+
+
xi=l x:,j=l
x:=,
3.1.6. Lemma. Let O : R3 + R~ be defined by @(x,y, r ) = (x, xy, y2, r). Let
f : R4 + R be a C m function. T h e n for almost all linear maps L : R4 + R the
map (f
L)@ has only non-degenerate critical points.
+
Proof. As in Lemma 3.1.5 assume that (f + L) o @ has a critical point along the
line (0,0,z). If this is degenerate, then each bi depends linearly on r . So the set of
linear maps L that yield degenerate critical points is of measure zero.
3.1.7. Lemma. Let S c {1,2, . . . , n). Let Ks be a compact subset of the open ball
Us in n S r . Suppose that f : U + R has only non-degenerate critical points on each
K s T h e n there is a positive number S such that if g : U + R is a C 2 function that
satisfies
I
Movie Moves
259
and
for i , j E S' at all points of K s , then gJUs has only non-degenerate critical points
on
Proof. See Lemma B in [9].
3.1.8. Lemma. Let I< be a compact subset o f R n + l . Let I<' be a compact subset
of Rn such that @(I<') C K where O ( x ,y, ( r ) ) = ( x ,x y , y 2 , ( z ) ) if n = 2 (resp. 3).
Suppose that f :
R is a C 2 function that has only non-degenerate critical
points in Ii', and suppose that f o O has only non-degenerate critical points in Ii".
Then the analogous conclusion of Lemma 3.1.7 holds.
-
Proof. Use the Taylor polynomial expansion for f and g .
3.1.9. Lemma. Suppose h : U -+ U' is a diffeomorphism from an open ball U that
contains 0 i n Rnfl. Suppose that h carries a compact set Ii's to the compact set
K $ . where K s = {i: E Ii' : xi = 0 for i E S ) , and Ii' is compact. Given an E > 0,
there is a 6 > 0 such that if a smooth map f : U -+ R satisfies:
1. If l < 6 ,
2. ) % ) < 6 ,
3.
f o r i = 1 , 2 ,..., n , and
a)
laXnaxj
<6, fori,j=
1 , 2 ,... , n
at all points i n K t , then for all S C { 1 , 2 , . . . ,n
+ 11, f o h satisfies:
1. (f0 h( < E ,
< E , f o r i E S ' , and
2.
3.
< L, f o r i , j
E St,
Proof. See Milnor [9] Lemma C.
3.1.10. Lemma. Let @ ( x ,y, ( 2 ) ) = ( x ,x y , y 2 , ( r ) ) if n = 2 (resp. 3). Let I< be a
compact neighborhood of 0 E Rn. Suppose h : U -+ U' is a difleomorphism from an
open ball U that contains 0 i n R ~ + ' .Suppose that h carries the compact set @ ( K )
to a compact set I<'. Given an E > 0 , there is a 6 > 0 such that if a smooth map
f : U + R satisfies:
1. I f l < 6 ,
2. ( z l < 6 , f o r i = 1 , 2 ,... , n + l , and
3.
I&[
<6, fori,j=
1,2 , . , n + 1
J . S. Carter €4 M. Sarto
260
at ali poznts zn I<', then f o h o Qr satzsjies:
1.
2.
3
If
11
< E,
< E , for i = 1 , .. . , n , and
oh001
1
E
, f
0
,
l
. . ,n,
at each poznt of I{.
Proof. See Milnor [9] Lemma C.
4. Fold Lines a n d t h e Self I n t e r s e c t i o n S e t s
It is known that any smooth map from an n-manifold (n > 1) to the plane is
approximated by a map only with two types of singularity: cusps and folds (see [7]
for definitions). The rank of the derivative of the map drops by 1 at a point on
these types of singularity. More specifically, the neighborhood of a point on the fold
is parametrized by (u, x, (y)) t, (u, x 2 ( fy2)) in case of n = 2 (resp. n = 3). The set
on which the rank drops is 1-dimensional and called a fold lane. A neighborhood
of a cusp is parametrized by (u, x, (y)) H (u,x3 ux(+y2)); CUSPS are isolated
+
Fig. 2. Fold lines
Movie Moves
261
CUSP
Fig. 3. A Cusp
Our goal in this section is to prove Theorem 4.1. We need this theorem for the
later sections. We briefly explain our motivations here. In order to show that the
set of movie moves is sufficient to express any isotopy between knotted surfaces,
we consider a 1-parameter family of "height" functions on the range. Theorem 4.1
allows us to assume that this family of height functions is as nice as possible. Notice
here that a critical point for one such height function extends to a 1-dimensional
set in the family (at least locally), and this forms a fold line. The image of this fold
in the projection gives rise t o those isotopy moves that involve folds.
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4.1. Theorem. Let n = 1 or 2. The set of functions q : Yn+'x I + R x I satisfying
the following properties i s open and dense i n the C2 topology.
(1) For ( n , k) = (1, I), (2, I ) , or (2,2), the singularities of the composition qk =
q o Q k consist only of folds and cusps.
(2) These singularities intersect the image of the multiple decker sets of @ transversely.
More precisely, Let Fk be the fold locus of q k . This is the submanifold of T@(k)
on which the rank of the derivative of qk drops by 1. Then Fk intersects the image
of Q,,k ( T @ ( rk))
, transversely.
4.1.1. Remarks. In case n = 1, the map Q has double and triple points. The
double point manifold in this case is 1-dimensional, and the triple point manifold
is 0-dimensional. Thus there are no folds along these strata. In case n = 2, the
map @ has double, triple, and quadruple points. The triple point manifold is 1dimensional and the quadruple point manifold consists of isolated points. These
self intersection sets do not have folds. Thus the fold lines exist in only the cases
when (n, k) = (1,I ) , (2,2), and ( 2 , l ) . The double point manifold may have cusps
on its boundary, as in Fig. 6.
Theorem 4.1 will be used to give a proof of the sufficiency of the Reidemeister
moves and A-type moves t o express any isotopy between knots in the smooth category (Sec. 5). By this theorem we can assume without loss of generality that any
isotopy between two knot projections can be approximated by an isotopy satisfying
the conditions in Theorem 4.1.
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262
J. S. Carter t3 M. Saito
Fig. 4. Moving crossing points past folds: unbranched
SCC
FIG
32
4
Fig. 5. Moving crossing points across folds: branched
-
-
-
Movie Moves
Fig. 6. Cusps on double point set
Fig. 7. Moving triple points across fold lines
263
264
J. S. Carter U M. Satto
In case n = 1, the singularities of q@ are fold lines and simple cusps. The fold
lines are properly embedded submanifolds of TB(l) that intersect the image under
0(2,1)
of the double decker manifold of a projection of a knot isotopy. The double
decker manifold is 1-dimensional, and when these pass over a fold, there is either a
branch point (type I) move, or one of the A-type moves (Fig. 14). The triple decker
manifold, in this case, consists of isolated points, and so by transversallty the fold
lines miss these points.
In case n = 2, then we use Theorem 4.1 t o prove that the movie moves suffice
to construct isotopies between knotted surfaces. In this case, @ is the projection t o
R3 x I of the knot isotopy. The singularities of q@ are isolated cusp points or
properly embedded fold lines. These intersect the image of the surface of double
decker points of F x I (where F is the knotted surface) in isolated points. Furthermore, the double point manifold will have simple cusps and folds, and these may
intersect the image of the triple decker manifold These intersections give rise to
the movie moves. Details are found In Sec. 7.
4.1.2. Proof of 4.1. The proof is analogous t o the proof of Theorem 3.1. We first
show that locally the fold lines intersect the multiple decker manifolds transversely
for an open dense subset of maps. Then we adjust the map, q, so that qQ,is nice on
bigger and bigger portions of the manifold X x I. T h a t is, in a neighborhood of q
there is a small open set of maps such that any map therein is nice on all of X x I.
The open set is the intersection of neighborhoods in which maps are subsequently
nicer on larger and larger portions of the manifolds. Thus the proof follows from
Lemmas 4.1.3 through 4.1.6.
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4.1.3. Lemma. Let n = 1 , 2 , let U be an open ball an Rn+', and let 0 5 r 5 n 2.
Let S c { I , . . . , r ) , and for i E S let U, = { I E Rn+' : t,= 0). The set of smooth
maps f : U -+ R2 such that f lU, has folds and sample cusps whzch zntersect the
remaznzng U3 and thezr zntersectzons transversely zs an open dense set zn the CCO
topology.
Proof. The map f, = f lU, has folds and simple cusps if it is 2-generic in the sense
of [7]. The szngular set, Sl(f,), is defined t o be the set of x E U,on which the rank
of the derivative drops by 1. Thus for most f,, the fold lines of f, = f lU, form a
submanifold of U,on which the rank of fz drops by 1, [7], p. 87.
For any i E S, the set of maps f such that f, has folds is open and dense.
Similarly, the set of maps that has folds on the intersection of these subsets is open
and dense. An argument about codimensions shows t h a t there are non-empty folds
only on the double point manifold when n = 2. (See [7], p. 143.) The intersection
of this finite collection of open dense subsets is also open and dense.
Now if a fold line does not intersect the image of the multiple decker manifold
transversely, or if it intersects at a cusp, then the submanifold of fold points can be
perturbed t o a transverse intersection. Moreover, there is a map that is close t o the
L
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original f , and the fold locus of the new map is the transversely intersecting fold.
This completes the proof.
-
= (x, xy, y2). T h e n for
R 2 , the composition LQ has folds containing (0,O).
4.1.4. L e m m a . Let O : R2 + R3 be defined by @(x,y)
almost all linear maps L : R3
Proof. The fold lines are the points at which the rank of DLQ drops by 1. But
the determinant of DLO can be written as Ax2 + By + Cy2. where the constants
A, B , C depend on the linear map L. For most values of A , B , C, det = 0 on a
quadratic curve containing (O,O), so for an open dense set of linear maps the folds
will be generic.
4.1.5. L e m m a . Let 0 : R3 --+ R4 be defined by @(x,y,z) = ( x , x y , y 2 , z ) . For most
linear maps L : R4 -+ R 2 , the fold locus of the map LO is a submanifold of R3.
Proof. This is a direct calculation.
4.1.6. L e m m a . Let O be the map defined in Lemma 4.1.4 or Lemma 4.1.5. Then
for most maps f from R3 (resp. R4) to R2, the folds of the composition fO intersect
the image of the double decker set transversely.
1
Proof. The fold lines are the points at which the derivative drops rank by 1, but
by the Lemmas 4.1.4 and 4.1.5 this happens for almost all linear maps and hence
for almost all maps f .
In this section, we include a set of illustrations depicting the types of cusps and
folds that arise for maps of 3-manifolds, and the intersection of these fold with the
higher multiplicity decker sets. These, the projections of the image of 0 , are the
Figs. 2 through 7.
5. T h e R e i d e m e i s t e r T h e o r e m a n d E l e m e n t a r y S t r i n g I n t e r a c t i o n s
In this section, we give a proof in the smooth category of the classical Reidemeister theorem in the case a height function is fixed during the isotopy. The elementary
string interactions are defined. And we show how to use Theorem 3.1 to decompose
knottings of surfaces.
5.1. Theorem. Suppose two classical link diagrams of isotopic links are given.
Fix a height function on the plane and assume that these diagrams have isolated
maximal, minimal, and crossing points. T h e n there exists a finite sequence of Reidemeister moves and A-type moves that transforms one of the diagrams into the
other. These Reidemeisler moves and the A-type moves are depicted in Figs. 10
through 14.
Proof. Let L denote a closed 1-manifold (finite collection of circles). Let H :
L x I --+ R3 x I denote an isotopy between knottings of L. Consider a projection
R~ + R, and let p : R3 x I --+ R2 x I denote this projection times the identity map
on I. The map p may be perturbed so that Q = pH is a proper general position
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map on the surface L x I. By Theorem 3.1, we may assume that the projection
q': R2 x I
I is everywhere Morse.
Furthermore, the projection q : R2 x I -+ R x I may be assumed to satisfy
the conclusion of Theorem 4.1. That is, when restricted to the image of a, the
projection q generically has folds and simple cusps, the fold lines intersect the
double decker manifold transversely (and hence in isolated points), and the only
folds on the double decker set are the branch points of the generic map a .
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Fig. 8. BirthIDeath
Fig. 9. Surgery
Fig. 10. Type I
Fig. 11. Type I1
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Fig. 12. Type I11
Fig. 13. Switch Back move
Fig. 14. Moving crossing across a fold
Consider the moves t o link diagrams that are depicted in Figs. 10 through 14.
In these pictures, the height direction - in which the top of the page is up is fixed. On the right side of each figure is a depiction of the projection p of the
corresponding move. The map q can be thought of as the projection t o the plane
of the picture.
A type I move (Fig. 10) is a critical point of the Morse function q1O2 on the
double point manifold. At this local optimum, the involution on the double decker
manifold has a fixed point. A type I1 move (Fig. 11) is a critical point of q1Q2 a t
which the involution on the double decker manifold is acting freely. A type 111 move
(Fig. 12) represents an isolated triple point of O .
Locally, the fold lines intersect the double point manifold transversely as in
Fig. 10 or 14. And if the fold line have critical points (with respect t o q'), then
there is a cusp singularity in the surface and this is depicted in Fig. 13.
The moves depicted exhaust the possibilities for critical points and transverse
intersections. This completes the proof.
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Note in the above proof that a branch point (resp. a maximal or minimal point
of the double point set, an isolated triple point) of the projection of an isotopy of a
classical link corresponds to the Reidemeister type I (resp. type 11, type 111) move
if the projection is a general position map and a height function on 3-space is fixed.
In general, for a general position map from a closed surface to 3-space, there may
also be maximal, saddle, and minimal points such that in a neighborhood of such a
point the surface is embedded.
Consider the intersection of the surface with a pair of planes on either side of
such a critical point where the planes are perpendicular to the fixed height function.
Then the curves of intersection differ by the birth of a small circle, the death of
a small circle, or a surgery between two arcs. These events correspond to a local
maximum, a local minimum, and a saddle point, respectively.
5.1.1. Definition. The elementary string interactions (ESIs) are birth, surgery,
death, Reidemeister type I, type 11, and type I11 moves. These are depicted in
Figs. 8 through 12, and they correspond to passing through critical points of a
surface or its singular set as described in the paragraph above.
5.2. T h e o r e m . Let h : F + R 4 be an embedded closed surface. Consider a
projection p : R4 + R3 to a subspace and a height function k : R~ + R. For almost
all projections p and height functions k , we get a projection p o f ( F ) such that (1)
p o f is a general position map, and (2) the slices of p o f ( F ) by parallel planes i n
R3 that are perpendicular to the height direction k gives a sequence of elementary
string interactions up to topological equivalence of link diagrams.
Proof. By Theorem 3.1 we have an everywhere Morse function on a projection of
F which we may assume is a general position map. Each of the elementary string
interactions represents a critical point of the surface, its double point manifold, or
a triple point. This completes the proof.
5.2.1. R e m a r k s . 1. If a height direction is also fixed in all of the 3-dimensional
cross sections, then one must include the A-type moves as elementary string interactions. There might be a good reason to do this from the point of view of constructing
invariants of knotted surfaces. But in this case, the list of movie moves becomes
2. Theorem 5.2 can be generalized to properly embedded compact surfaces in
R 3 x I.
3. Theorem 5.2 can also be generalized to generically mapped surfaces in 3manifolds.
6. Roseman's T h e o r e m
6.1. T h e o r e m ( R o s e m a n ) . Suppose two projections in 3-space of isotopic knotted
surfaces in 4-space are given. Assume that they are the images of general position
maps. A set of moves between generically mapped surfaces are depicted in Figs. 15
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through 22. Then, there exists a finite sequence of moves from this set which transf o m s one of the projections t o the other up t o topological equivalence.
Fig. 15. An optimum on the surface of self intersections with fixed points
Fig. 16. A saddle on the surface of self intersections with fixed points
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f
st-tF/G
zg
Fig. 17. An optimum on the surface of self intersection with no fixed points
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Fig. 18. A saddle on the surface of self intersections with no fixed points
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Fig. 19. An optimum on the triple point set with no fixed points
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Fig. 20. Moving a branch point through a triple point
Fig. 21. Mixing type I1 and type I deaths on the double decker set
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SEE
JSC
Fig. 22. A quadruple point or tetrahedral move
R3 denote
Proof. Let H : F x I -+ R4 x I denote an isotopy. Let p : R4
a projection. We may choose the direction of p such that the composition O =
p ( H x id) is a generic map of the 3-manifold F x I. Let q : R~ x I --t I denote
the projection. This may be chosen so that q is everywhere Morse on O ( F x I).
Without loss of generality these choices can be done preserving the property that
H is a n isotopy since isotopies form an open dense set (see [13]).
We proceed to analyze the critical points of qOz, q03, and qQ4. (The composition qO has no critical points because H is an isotopy, and in perturbing q to be
Morse, we did so in such a way that the property of being an isotopy is preserved.)
Consider the critical points of the double decker manifold. Recall from remark 3.1.1 that since q is Morse on the double point manifold, it is equivariantly
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Morse on the double decker manifold. The double point manifold of a generically
mapped 3-manifold is a surface. And critical points of surfaces come in two flavors:
Optima (maxima or minima) and saddles. We think of the former as a quadratic
cone and the latter as a saddle.
Since there is an involution on the double decker manifold there is either a line
of fixed points on the saddle or quadratic cone, or there are a pair of saddles or
cones that are interchanged. The line of fixed points can be any axis of symmetry.
In Fig. 15 the move that corresponds to an optimum with a line of fixed points
is depicted. Figure 17 depicts the move that corresponds to a free action on the
quadratic cones. A saddle of double decker points with a line of fixed points is
depicted in Fig. 16. In Fig. 18 there is a pair of saddles that are interchanged by
the involution.
Next we consider the critical points on the triple decker set. The triple decker
manifold is a collection of curves immersed in F x I as the set of double points of
the double decker manifold. Morse critical points of curves are optima. Since these
curves are the double points of an immersed surface, they are the trace of either
type I1 moves or type I moves. Here the remark 5.2.1 (2) is being applied to the
double decker set which is an immersed surface.
In case three disjoint type I1 moves occur on the double decker surface, we have
two triple points of the knot projection cancelling. This is depicted in Fig. 19.
Locally, at a triple point of a surface, 3 arcs of double points intersect. If an
arc at a triple point returns to the triple point in an a fashion, then the arc of
double points with which the alpha is involved is embedded in F. (This matter is
illustrated in detail in [I].) Thus, if a triple point dies in the isotopy via a type I
move in the double decker set, there is a type I1 move on the companion sheet.
Thus the illustration given in Fig. 21 gives the only possible way a type I death of
a triple point can occur.
This projects to the Fig. 20.
Finally, the map can have quadruple points; such points are isolated. The corresponding move is depicted in Fig. 22. It represents a version of the Zamolodchikov
equation [14].
Since all possible Cg equivariant critical points of surfaces, all possible Cg
equivariant critical points of curves, and all possible C4 equivariant actions on 4
points have been listed, this completes the proof.
7. Movie Moves
In this, our final section, we list moves to movies that are sufficient to construct
any isotopy of knotted surfaces. These moves were announced in [2].
Let h : F -. R 4 be an embedded closed surface (i.e., a knotted surface or a
knotting). Without loss of generality we may assume that p o h is a general position
map where p : R 4 -. R3 is a projection. Furthermore from Theorem 5.2 we may
assume that p o h is a sequence of ESIs when we fix a height function on 3-space
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and take slices by parallel planes perpendicular to this fixed height direction. Such
a projection p o h ( F ) with a fixed height function on 3-space is called a movie
description (or simply a movie) of the knotted surface. The slices described above
are also called a movie. Two such projections are topologically equivalent if there
is an isotopy between them sending the double point set and ESIs of one t o those
of the other. When we take slices, a set of moves between sequences of ESIs are
depicted in Figs. 23 through 37. These are called movie moves. The goal of this
section is to prove that any two projections of isotopic knotted surfaces are related
by a sequence of these moves.
Each movie move corresponds to one of the following:
1. A critical p o i n t of the self intersection set of t h e k n o t projection.
These are movie parametrizations of Roseman's original seven moves, where
we interpret an isolated quadruple point as a critical point.
2. Passing the surface of d o u b l e p o i n t s transversely t h r o u g h a fold
line. At most, those moves that correspond to the unbranched moves illustrated in Fig. 4, and the top two moves illustrated in Fig. 5 are necessary.
3. Passing a t r i p l e p o i n t transversely t h r o u g h a fold line of the d o u b l e
decker set. The projection of this move is illustrated in Fig. 7.
4. Cusps. There is a cusp on the fold line that corresponds t o the cancellation
of a birthldeath and a surgery. There are also two types of cusps on the
surface of self intersections. These are illustrated in projection in Fig. 6.
Some comments on movie moves follow. Note from Theorem 3.1 that any movie
is described by a sequence of elementary string interactions. These correspond
crossings and maximal/minimal points in classical case. A movie of a knot that is
given as a sequence of elementary string interactions can be transformed to another
by a sequence of the moves to movies that are depicted. Thus the movie moves are
elementary moves t o sequences of elementary string interactions that are sufficient
to construct any isotopy.
In the depiction of the movie moves we have adopted the standard convention
of listing movie moves with only one possible set of crossing data. We leave it
as an exercise for the reader to find all the crossings for which the movies moves
remain valid. Furthermore, the movie moves can be read up or down. We have only
depicted maxima, but minima occur upon turning the movie move upside down.
Moves in which critical points can change level are not listed. These are analogous to the braid relations uiuj = uj ui if li - jl > 1. One such move is used in
Fig. 38.
7.1. T h e o r e m . Suppose two projections in 3-space of isotopic knotted surfaces
in 4-space are given. W e may assume that they are described as sequences of ESIs
with respect to a fixed height function on 3-space. T h e n there exists a finite sequence
from the set of movie moves depicted i n Figs. 23 through 3 7 which transforms one
to the other up t o topological equivalence.
i
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Fig. 23. Removing/adding branch points through an optimum on the double decker set
Fig. 24. Removing/adding branch points through a saddle in the double decker set
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Fig. 25. Removingladding circles of double points through an optimum
Fig. 26. A saddle move on the double decker set
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Fig. 27. Removing/adding a pair of triple points through an optimum on triple point manifold
Fig. 28. Moving a branch point past a sheet via an optimum on the triple point manifold
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Fig. 29. The Zamolodchikov Move: a quadruple point of the isotopy
Movie Movea
Fig. 30. A cusp in the isotopy
Fig. 31. Moving a branch point through an optimum fold
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Fig. 32. Moving a branch point through a saddle
Fig. 33. Moving a plane past an optimum fold
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Fig. 34. Moving a plane through a saddle fold
Fig. 35. A cusp of the double point manifold
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Fig. 36. A cusp of the double point manifold that involves a branch point
Fig. 37. Moving a triple point past a fold in the double point manifold
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Fig. 38. Factoring the unusual moves of branch points past a saddle
Proof. By Theorem 3.1 and 4.1 we may assume that the projections of the isotopy
are everywhere Morse and fold lines intersect the multiple points transversely. The
Roseman moves exhaust the set of Morse critical points that can occur on the
multiple point sets. And the list of folds intersecting the multiple point set is
complete listing of these transverse intersection.
All that remains t o be shown is that the trace of the moves that are illustrated
in the bottom frame of Fig. 5 can be factored into other moves. This factorization
is given in Fig. 38. The third frame of Fig. 5 can clearly be written as a cusp on the
double point set (Fig. 36 and Fig. 6).
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Acknowledgements
Jose Barrionuevo helped us with proving Morse functions are dense on neighborhoods of branch points. We have had valuable conversations with Cameron Gordon,
Kunio Murasugi, Dan Silver, arid Richard Hitt. Jacob Towber and John Fischer
impressed upon us the urgency of writing down these results. We have received
encouragements from L. H. Kauffman and J . D. Stasheff throughout the period we
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have been engaged in this and related research. We especially thank Dennis Roseman for giving us permission to give a proof of his work. Many of the ideas for this
paper are found in his manuscripts. Although we have not explicitly listed the work
[5] as a reference, we drew upon it at several points to better understand folds and
cusps. Yuuichiro Sugimoto gave a careful reading of an earlier manuscript, and we
benefited from his suggestions. Finally we thank the referee for valuable comments.
References
[ I ] J. Scott Carter, Extending immersions of curves t o properly immersed surfaces,
Topology and its Applications 40 ( 1 9 9 1 ) , 287-306.
[2] J. Scott Carter and Masahico Saito, Syzygies among elementary string interactions
i n dimension 2 1 , Lett. in Math. Phys. 23 ( 1 9 9 1 ) , 287-300.
[3] Fischer Jr. and E. John, Braided tensor 2-categories and 2-knots, Preprint.
[4] Igor Frenkel and Gregory Moore, Simplex equations and their solutions, Comm. Math.
Phys. 138 ( 1 9 9 1 ) , 259-271.
[5] George K . Francis, A Topological Picturebook, Springer Verlag, New York ( 1 9 8 7 ) .
[6] Peter J. Freyd and David Yetter, Braided compact categories with applications t o low
dimensional topology, Advances in Math. 77 ( 1 9 8 9 ) , 156-182.
[7] Martin Golubitsky and Victor Guillemin, Stable Mappings and T h e i r Singularities,
GTM Vol. 14, Springer Verlag, New York (1973).
[8] Tatsuo Homma and Teruo Nagase, O n elementary deformations of the m a p s of surfaces i n t o 3-manifolds I, Yokohama Math. J. 33 ( 1 9 8 5 ) , 103-119.
[9] John Milnor, Lectures o n the h-Cobordism Theorem, Mathematical Notes, Vol. 1 ,
Princeton University Press, Princeton, NJ (1965).
[ l o ] Kurt Reidemeister, Knotentheorie, Springer Verlag, Berlin (1932). K n o t Theory, BSC
Associates, Moscow, Idaho (1983).
[ l l ] Dennis Roseman, Reidemeister-type moves for surfaces i n four dimensional space,
Preprint.
[12] Dennis Roseman, Projections of codimension 2 embeddings, Preprint.
[13] Dennis Roseman, Elementary moves for higher dimensional knots, Preprint.
[14] A. B. Zamolodchikov, Tetrahedron equations and the relativistic S - m a t r i x of straightstrings i n 2 1-dimensions, Commun. Math. Phys. 79 ( 1 9 8 1 ) , 489-505, reprinted in
Yang-Baxter Equation i n Integrable Systems, ed. Jimbo, World Scientific Publishing
Co., Singapore (1989).
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