Computer study of knots

COMPUTER GRAPHICS AND IMAGE PROCESSING

11, 150-161 (1979)

Computer Study of Knots* S. D. SUPANEKAR

The Bombay Textile Research Association, Ghatkopar, Bombay 400 086, India Received April 5, 1979; accepted May 14, 1979 We define the Tait representation for a knot and determine its variants, which occur because of alternative choices of the arbitrary elements. We then note that any knot can be built up from six unit operations and consider how its Tait representation can be obtained as computer output when the knot is input graphically. This opens up an avenue for deciding whether or not two prime knots belong to the same knot type. Finally we explain how a FLASH knot (projection) can be generated by the computer on the receipt of its Tait representation. 1. INTRODUCTION K n o t theory is a b o u t 100 years old and has an impressive literature. An account of the theory at an intermediate level of difficulty is given by Crowell and Fox [-1]. T h e studies in the past confine themselves to p r o b l e m s of m a t h e m a t i c a l interest alone. However, with the a p p e a r a n c e and p o p u l a r i t y of w a r p k n i t t e d fabrics such studies have a potential application in developing new cloth constructions. Naturally, they will be a c c o m p a n i e d b y a shift in emphasis and methods. C o m p u t e r s have been used in k n o t and link studies for deriving their algebraic invariants such as Alexander polynomials from different schemes of k n o t specification. (See [-21 and the references cited therein.) B u t there is no reported work on obtaining the representation itself f r o m the c o m p u t e r when the graphic input is the knot. I n this p a p e r we consider one f o r m of k n o t specification, the T a i t representation, which is obtained in this way. We s t u d y the v a r i a t i o n s arising out of alternative choices of a r b i t r a r y elements and because of the transformations ~1, ft2, ft~ [-3, 4]. W e t h e n t a k e u p the p r o b l e m of obtaining the (unique) T a i t representation of a k n o t when the latter is built up step by step on a graphic input terminal of a computer. C o m b i n i n g these we are enabled to c o m p a r e two oriented knots. T h u s this novel application of c o m p u t e r graphics m a k e s it possible to decide directly whether two k n o t diagrams are equivalent. This problem cannot be settled b y topological m e t h o d s [-1, p. 163]. Finally, we also deal with a partial converse of the earlier p r o b l e m : to reconstruct a k n o t projec* This paper is a revised and extended version of an earlier paper presented at the Bangalore Convention of the Computer Society of India in January 1979. 150 0146-664X/79/I00150-12502.00/0 Copyright ~ 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.

COMPUTER STUDY OF KNOTS

t

151

6

FIG. 1. False Lover's Knot in a stylized form. tion from its Tait representation, restricting ourselves to a class of knots. We can thus conclude t h a t computer graphics is a v e r y effective medium for experimentation in knot studies. 2. THE TAIT REPRESENTATION Let us begin with a few useful conventions. In our style of presenting a k n o t (projection) there will only be straight s e g m e n t s - - e i t h e r horizontal or vertical. Further, the crossings will not occur on vertical segments nor at the corners. Given a knot, we fix on it, arbitrarily, a point 0 called the initial point and a positive orientation shown by arrows. T h e Tait representation depends on these as well as on the side from which the knot is viewed. At a crossing we also speak of the u p p e r and lower nodes [-2] or of over-and-under crossings. Suppose t h a t the knot has N crossings; let XN

=

{1, 2, . . . , N}

YN = Xyk.J{O}

and and

X~- = {-1, -2, ..., -N}, ZN = X N U X N - .

Let the 2N nodes of the knot be numbered serially starting from 0 and following the positive orientation. We define two maps U : XN -~ X2x and V : XN -~ X2N such t h a t U ( I ) ( V ( J ) ) denotes the serial n u m b e r of the I t h (Jth) undercrossing (overcrossing). As these maps are injective their ranges partition X~N into two sets of equal size. As usual, the inverse maps U -1 and V -1 are defined from these ranges and are bijections. We label the crossings by the values of U (I) and also the u p p e r node at each crossing by the same number. (This arrangement works quite satisfactorily as we shall notice in the sequel.) The lower node receives the label - U ( I ) . In this scheme every node has a serial n u m b e r and a label. T h e natural bijection T: X2N ~ ZN between these is the Tait sequence of the knot. B y way of illustration the Tait sequence of the knot depicted in Fig. 1 is --1, 7, --2, 5, --3, 6, --4, 8, --5, 3, --6, 1, --7, 2, --8, 4. In a few simple cases the Tait sequence determines the knot. However, in the general situation we must specify, additionally, the sense of rotation from underto-over crossing [-3, p. 18], at every double point of the knot. We therefore define a mapping W : XN ~ Z1 which takes the value 1 ( - 1) at the crossing labeled I if the rotation is counterclockwise (clockwise), 1 ~4 I ~< N.

152

S. D. SUPANEKAR

The sequence pair (T, W) constitutes the Tait representation of the knot. It is similar to Penney's E5] word representation except t h a t there is a natural order in the subsequence of its negative terms. Other modes of knot representation exist F6-9J ; but the Tait representation is simple and readily amenable to machine manipulation. We will use it exclusively in the sequel. There exists a somewhat inelegant way of combining the two mappings into a single mapping without information loss. Define U': XN -~ X~N SO t h a t for the I t h undercrossing U'(I) = 21 - 1 if W ( I ) = - 1 , if W ( I ) = 1.

= 21

The other details are similar to the case of the Tait sequence. The natural injection 0: X2N -+ Z2N specifies the knot completely. If we identify the two nodes at a crossing, a knot projection becomes a planar, directed graph with crossings as its vertices. Its arcs m a y be specified by the pairs of consecutive terms T ( I ) T ( I ~- 1) of the Tait sequence. (We identify T ( 2 N + 1) with T(1).) The associated undirected graph is also a planar, connected, regular graph with the degree of each vertex equal to 4. By Konig's theorem the chromatic number of a knot map is 2. This fact is used in one of the knot representations E3, p. 201. The associated undirected graph, called a normal mapping, may be denoted by S ( I ) = ]T(I)I, 1 <~ I <~ 2N, and its edges by S ( I ) S ( I - t - 1) rl0~. A necessary and sufficient condition for a plane curve to be a normal mapping is given by Marx [111. 3. VARIANTS OF THE TAIT REPRESENTATION In this section we consider the distinct Tait representations of a knot t h a t occur with alternative choices of initial points, orientations, or viewing side. (A) Shift of the initial point 0 in the positive sense skipping one crossing is described by the operator ~:

o~(T(I)) = T ( I + 1)

if V(1) = 1, 1 ~< I < 2N,

a ( T ( 2 N ) ) = T(1). Otherwise, the action of a is described by the permutation //

-N

-(N-i)

....

2

-1

1

2

...

N-

-(N-l)

-(N-2)

....

1

-N

N

1

....

N-2

1

N N-

~. 1]

The action of a on the weight sequence W must also be prescribed. We observe t h a t although the label of a crossing m a y undergo a change its weight does not. (This remark also applies in cases (B) and (C) given below.) Hence

a(W(a(T(I)))) = W(T(I)),

1 ~ T(I) <~ N.

We have O/2N =

Identity.

(B) Inversion without change in the first undercrossing or viewing side is

COMPUTER STUDY OF KNOTS

A GO a

C

I°

CD

B

MOVE b

A

B ~ D HOP d

CRAWL c

B

153

E

E

I

A

XB

F FORD e

C

JUMP f

FIG. 2. Unit operations on a knot. described by the operator/3 which is a p e r m u t a t i o n of ZN and a reversal of terms ; (-N --2

-(N-l)

....

2

-1

1

2

..-

N-1

--3

....

N

-2

1

N

-..

3

N). 2

T h e action of ~(~) on W is given by the same formula as t h a t above with a replaced by ~ (~,). Again ~ = Identity. (C) Viewing from the opposite side is represented by the transformation ~, when there is no change of the initial point or inversion. Unlike a and/~, 7 reverses the signs of the entire Tait sequence. We have ~(T(V(I))) = -I, " y ( T ( U ( I ) ) ) = I,

and 72 = Identity. T h e transformations a, /3, ~, generate a group Go [-12] which is a subgroup of Ga to be described later. Clearly the order of Go is 8N. 4. TAIT REPRESENTATION FROM THE KNOT We consider the problem of obtaining the Tait representation of a knot as the machine o u t p u t on inputting the knot graphically. This work was carried out on a Tektronix 4012 terminal using a DEC-1077 system. T h e input tools used were the cross-hair cursor and the terminal k e y b o a r d ; the procedure is explained below. In our style of presentation we find t h a t any knot can be built up from a set of six unit operations. T h e y are enumerated below. (1) Drawing a horizontal or vertical segment (Fig. 2a). We describe this operation b y " G O . "

154

S. D. SUPANEKAR

(2) Moving from position A to position B without drawing the connecting segment (Fig. 2b). We call it " M O V E . " (3) Creating a new overcrossing (Fig. 2c). T h e upper node is at B and is shown by a cross. It is called " C R A W L . " (4) Creating a new undercrossing (Fig. 2d). T h e lower node is shown by the dot at C. This is called " H O P . " (5) Negotiating an existing undercrossing (Fig. 2e). The crossing occurs at C and the operation is called " F O R D . " (6) Jumping an existing overcrossing (Fig. 2f). The crossing occurs at B. We call it " J U M P . " T h e commands to execute these operations are also designated by the same calls. As a m a t t e r of working detail, it is convenient with respect to operations (3) and (4) to draw a circle about the new node. The radius of the circle is u n i m p o r t a n t so long as two such circles do not intersect. At the time of c o m p l e m e n t a r y recrossing the leading end of the knot is brought to a point such as P within this circle (Fig. 3a). T h e command F O R D or J U M P is issued after this. T h e p r o g r a m discovers the center of the circle and effects a linear displacement along the diameter through P to the point Q (Fig. 3b). T h e segment PQ m a y or m a y not be drawn as per command. While the knot is built up step by step t h r o u g h the applications of unit operations, its Tait representation develops concurrently. For the Tait sequence we introduce three integer variables J, K, L which are initialized to zero but which range over sets XN, X2y, and ZN during the program execution. We also have an integer array T A I T (K) giving the sequence and two real arrays SX(J) and SY(J) which store the coordinates of the crossings. For the determination of the weight sequence W we note the coordinates XC, YC of the current beam position after every m o v e m e n t ; we also store the coordinates XP, YP of the previous beam position. These two pairs enable us to know the displacement directions D X and D Y along the horizontal and vertical axes. B o t h these are maps from XN to ZI. In our scheme of presentation the former must be stored in an array DX(K). T h e value of W at a crossing is determined at the time of recrossing from DX, D Y and the n a t u r e of t h a t crossing. Thus, e.g., if D X = D Y = 1 and the recrossing at J is effected by a c o m m a n d F O R D then W(J) = 1, etc. Table 1 summarizes the actions following each command.

FIG. 3. Alignment at complementary recrossing.

Linear displacement A B only

Marker ( X ) and circle drawn

Marker ( • ) and circle drawn

Segment FG drawn through the node; circle erased

Segment D E drawn jumping the node; circle erased

MOVE

CRAWL

HOP

FORD

JUMP

K -* K + 1 L --* L + 1

K --* K -b 1

J --~ J + 1 K--* K-]- I L---*L+I

J --~ J -b 1 K--~K--kl

.

.

Increments

.

.

.

.

Arrays S X , S Y , and D X

Arrays S X and S Y scanned; nearest crossing ( = J P ) and corresponding serial n u m b e r (~-KP) determined

(KP)

TAIT(K) = -L; T A I T (KP) = L

= -TAIT

T A I T (K)

Arrays S X and S Y scanned; nearest crossing ( = J P ) and corresponding serial n u m b e r ( = K P ) determined

--

T A I T (K) = - L

.

.

Tait sequence

Coordinates of C and D X (K) stored

Coordinates of B and D X (K) stored

The references to points in columns 2 and 4 refer to Fig. 2.

Linear displacement, connect . ing segment drawn

Display

GO

Command

Buildup of Tait Representation with Knot"

TABLE 1

W(JP) = -DX(KP)

X DY

W(JP) = DX(KP) X DY

Weight sequence

¢j1

©

©

©

156

S. D. SUPANEKAR

o

b

1

FIG. 4. Operation ~l; removal of a loop. 5. COMPARISON OF KNOTS The methods developed above muy be used to carry out a machine comparison of two prime knots. For this purpose we must evaluate the effects of operations f~1, f~, and f~3 ~3, pp. 7-81 on the T a i t representation. Operation f~l removes a loop from the knot (Figs. 4a and b). T h e presence of such a loop is revealed by a pair of two consecutive terms of the T a i t sequence which differ in sign only, e.g., - K , K. T h e i r removal will result in a new sequence which is shorter by two terms. Moreover, it is necessary to change the labels of crossings following the eliminated one if there are any, and to "close the g a p " in the sequence. The new values of the altered terms are given by the transformation ~ defined by ~(T) --- T - - 1 if T > K, = T+

= T

1

if T <

--K,

otherwise.

T h e order of the terms in the new sequence does not change, although the rank decreases for the terms following the removed pair. The length of the weight sequence is shortened by 1 and the previous remark applies here as well. Operation e2 removes a crossing pair (Figs. 5a and b). I t can be recognized by the existence of two pairs consisting of the same consecutive pair of integers differing in sign only; e.g., ...,-K,

-(K+

1), . . . , K , K

+ 1, . . . ,

or

....

K+I,K

....

,-K,-(K

+ 1), . . . ,

etc. The elimination of such a crossing pair diminishes the length of the T a i t sequence by 4, the order of the terms remaining unaffected. T h e operator specifying the change is defined by , ( T ) = T -- 2,

if T >

( K - k 1),

= T+2,

if T < - ( K q -

=T

otherwise.

1),

v(W) merely removes two terms from the sequence and closes the gap.

.J. a

b

FIG. 5. Operation f~,; removal of a crossing pair.

COMPUTER STUDY OF KNOTS

157

b

Fro. 6. Operation e3; deformation of a triangle. Unlike fh and ~2, operation f13 does not shorten the Tait sequence as it merely deforms one triangle of arcs into another (Figs. 6a and b). Such a triangle can be detected in the Tait sequence b y the presence of three pairs of terms such as ...,L,K,

..., --K, --(K+

1), . . . , K +

1, - L . . . . ,

etc. T h e order of the t e r m pairs in the sequence as well as t h a t of individual terms in the pairs containing at least one positive term can be arbitrary. T h e operator ~ which describes the change acts as follows: In the Tait sequence the term pairs consisting of at least one positive terms, i.e., L, K or --L, K + 1 interchange K with K + 1 and also the order of terms in each pair. Further, (a) If K and K + 1 are both first or both second terms in the respective pairs then ~" leaves W unchanged; and (b) If K and K + 1 occur at different places in these pairs ~ changes the signs of W(K) and W(K + 1). B y repeated application of al and ft2 the n u m b e r of terms in the Tait representation can be reduced to a minimum. We then run another cheek on the T a i t sequence. This consists of operating on it by a t, 1 ~< I < 2N, and finding whether any of the resulting sequences contains a subsequence T' of consecutive terms such t h a t M belongs to T' if and only if - 3 I belongs to it. If the Tait sequence contains no such subsequenee T' t h e n it is called prime [-131. Suppose now t h a t we have two prime knots ~ and X, each represented by a prime sequence. We can determine whether or not they belong to the same knot t y p e by subjecting one of the representations to the elements of Go or ~"or combinations of these. If the two representations m a t c h for some transformation then they belong to the same knot type. If a prime knot has N crossings and A deformable triangles, then it will have 8N X 2 a Tait representations. These form the group Ga which contains Go as a subgroup. It m a y be observed that the Tait representation is a stronger invariant of a knot than its group. For instance, it is well known [-1, p. 1313, t h a t the groups of G r a n n y and square knots are isomorphic. B u t their Tait sequences are distinct ; because the G r a n n y knot is alternating, f is not applicable to its Tait representation, while under a, ~, and %,the signs of the terms of the Tait sequence remain + and -- alternately. This does not happen with the square knot.

158

S. D. SUPANEKAR

10 i

.[

:

6

Fin. 7. Necktie knot--non-FLASH. 6.

KNOT RECONSTRUCTION

We next come to the more difficult converse p r o b l e m of k n o t reconstruction. H e r e the solution is obtained only for a class of knots which we will call F L A S H knots. These are characterized by the fact t h a t the associated sequence S(I), 1 ~< I ~ 2N, contains a run of the futl set XN, t h o u g h not in the natural order. I f we a p p l y the t r a n s f o r m a t i o n a repeatedly to the T a i t sequence of such a k n o t and examine the associated sequence on each occasion, we find t h a t at a certain stage the latter contains a p e r m u t a t i o n of XN in each of its halves. W h e n this happens the First Loop in the k n o t Appears in the Second H a l f of the sequence, i.e., at the (N A- 1)st term. T h e word F L A S H is an a c r o n y m derived f r o m this situation. A consequence of the same fact is t h a t all the crossings of the k n o t can be m a d e to occur on a single line, the base line. Figure 1 represents a F L A S H and Fig. 7 a n o n - F L A S H , respectively. T h e conditions necessary for a sequence to be a T a i t sequence are: (1) I t should have an even n u m b e r (say 2N) of t e r m s ; (2) it should be a bijection with ZN; and (3) the difference T -1 ( - - T ( I ) ) T -1 ( - - T ( I A-1))

should

be odd for

N <<.I < 2 N . T h e last condition ensures t h a t the n u m b e r of crossings spanned b y the arc

T ( I ) T ( I -4- 1) is even so t h a t if the leading end of the k n o t enters a loop, there is an exit available for it. I n our style of k n o t presentation, the completion of the d i a g r a m depends u p o n our making correct choices of direction after an arc is drawn. This point is borne out by Fig. 8. C o m p a r i n g it with Fig. 1 we find t h a t a choice has to be exercised impasse 4o

h

,t

FIG. 8. Occurrence of impasse.

:, 8~(

COMPUTER STUDY OF KNOTS

a

159

b

c

d

Fro. 9. Equivalent ways of joining two crossings. after the recrossing at 6 is made. The choice of Fig. 8 leads to an impasse while t h a t of Fig. 1 enables us to complete the knot. Two conventions explained below help to shorten the program. These concern equivalent ways of joining two crossings. Suppose t h a t we have to join crossings A and B from the same side of the base line. This can be realized in two ways as shown in Figs. 9a and b. T h a t they are equivalent can be seen by imagining the portion of the loop surrounding the base line in Fig. 9b to be thrown over onto the other side of the base line and pulled in. This act will result in the joining of the two crossings as shown in Fig. 9a. In the event of multiple joinings of this kind we can repeat the operation from the outermost to the innermost loops. Likewise, there is a choice in joining two crossings from opposite sides, as shown in Figs. 9c and d. Here we will choose the style of Fig. 9d. The program first reduces the knot to one with a minimum n u m b e r of crossings, and then checks whether it is a F L A S H knot. T h e next step is to construct the base line on which all crossings appear. Figure 10 illustrates the initial steps for the "False Lover's K n o t . " In F L A S H knots one loop is formed at every t e r m in the second half. T h e first loop, which we always complete from below, is shown in Fig. 10b. Next, we define a new function L S : X N ---> YN b y LS(I) = [T-I(--T(I

3 - N ) ) -- T - I ( - T ( I

1))l;

3- N - -

I > 1

this function gives the n u m b e r of crossings spanned by the I t h loop. Also, e.g., I

~7

×5

o2

e3

.,6

5 o~

6 X J"

_o4

Q I O-

? X

5t X

2 ~ O

i,-

•

4

8 X

b

t,, •

X

r---

- -

4

c

Fro. 10. Initial steps in the construction of a False Lover's Knot.

160

S. D. SUPANEKAR

in Fig. 10b, LS(1) = 4. It is useful to determine the height of a loop which must accommodate subsequent loop nests. As the knot construction progresses the knot map is balkanized. We need to label the various regions t h a t are formed. Initially, the entire region of the plane is labeled 0. After the completion of the first loop the region below the crossings 3, 4, 6, 8 is distinguished by labeling it 1. The other region continues to have its previous label. The point is t h a t the regions above or below a crossing acquire different labels as the construction proceeds. Eventually, once the leading end of the knot makes the recrossing, the regions above or below it form part of the knot. We then label them --1, which remains fixed thenceforth. These concepts are utilized by defining two integer-valued functions A (J, L), B (J, L) giving the labels of the regions above and below the crossing with serial number J immediately after the L t h loop is completed. To illustrate, in Fig. 10a,

A(J,O) = B(J,O) = O,

l ~ J ~ 8;

A(J, A(4, B(J, B(J,

1) 1) 1) 1)

= = = =

0, B(4, 1) = --1, 0, 1,

J ~ 4,

A(J, B(J, A(J, A(J, B(J,

2) 2) 2) 2) 2)

= -= = =

0, 0, B(J, 2) = --1, 0, 1,

1 1 4 6 6

in Fig. 10b,

1 ~< J ~4 3, 5 ~< J ~ 8;

in Fig. 10e, ~ J ~ J ~ J ~< J ~ J

~ 3, ~ 3, ~< 5, ~4 8, ~ 8;

etc. We also define a function E(L) which signifies the label of the region where the leading end of the knot lies immediately after loop L is completed. In the above illustration E(1) = 0, E(2) = 1, etc. Finally, we define three more functions A X ( L ) , A Y ( L ) , and B X ( L ) from XN to Z~; the first two of these specify the approach parallel and perpendicular to the base line at the time the L t h 10op is completed. Figure 11 illustrates these functions. The third function B X ( L ) describes the direction taken by the leading end parallel to the base line after the L t h loop is completed. For example, in Fig. 1, BX(3) = - 1 and this enables us to complete the knot, whereas BX(3) = 1 leads to an impasse.

"q t

a AX(L) = - I AY(L)=-I

t b

AX[L)=+I .a,y(L) =-I

_.

f

t

t

L..

c

AX(L)=+I AY(L)=+I

FIG. 11. Approach functions AX and A Y.

d

AX (L)=-I AY ( L)=+I

COMPUTER STUDY OF KNOTS

161

W e u t i l i z e t h e g i v e n w e i g h t s e q u e n c e W in t h e e v e n t A ( J , L) = B ( J , L) = E ( L ) in o r d e r to m a k e a choice in j o i n i n g t h e crossings. T h e f u n c t i o n s d e f i n e d a b o v e e n a b l e us to c o n s t r u c t F L A S H k n o t s . O n c e t h e distance between consecutive points, the gap at an undercrossing, and the minim u m h e i g h t of a loop a r e specified t h e y c a n be u s e d to g e n e r a t e t h e c o o r d i n a t e s of p o i n t s which, w h e n c o n n e c t e d s e q u e n t i a l l y , d i s p l a y t h e k n o t . REFERENCES 1. R. C. Crowell and R. H. Fox, Introduction to Knot Theory, Springer-Verlag, New York, 1977. 2. H. F. Trotter, Computations in knot theory, in Computational Problems in Abstract Algebra (J. Leech, Ed.), pp. 359-364, Pergamon, London, 1970. 3. K. Reidemeister, Knotentheorie (Reprint), Chelsea, New York, 1948. 4. J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30, 1928, 275-306. 5. D. E. Penney, Establishing isomorphisms between prime tame knots in E~, Pacific J. Math. 40, 1951, 675-680. 6. R. Riley, Homomorphisms of knot groups on finite groups, Math. Comp. 25, 1971, 603-619. 7. J. H. Conway, An enumeration of knots and links and some of their algebraic properties, in Computational Problems of Abstract Algebra (J. Leech Ed.), pp. 329-358, Pergamon, London, 1970. 8. E. V. Krishnamurthy and S. K. Sen, Algorithmic line notation for the representation of knots, Proc. Indian Acad. Sci. Sect. A 67, 1973, 51-61. 9. J. H. Conway and C. McA. Gordon, A group to classify knots, Bull. London Math. Soc. 7, 1975, 84-86. 10. S. MacLane, A combinatorial condition for planar graphs, Fundam. Math. 28, 1937, 22-32. 11. M. L. Marx, The Gauss realizability theorem, Proc. Amer. Math. Soc., 22, 1969, 610-613. 12. J. W. Alexander and G. B. Briggs, On types of knotted curves, Ann. of Math. 28, 1926/1927, 562-586. 13. J. R. Martin, Determining knot types from the diagrams of knots, Pacific J. Math. $1, 1974, 241-249.