Explorations into Knot Theory: Colorability
Rex Butler, Aaron Cohen, Matt Dalton, Lars Louder,
Ryan Rettb erg, and Allen Whitt
July 25, 2001
Abstract
We explore prop erties of a generalized \coloring" of a knot including
existence, changes in regards to satellite knots and sp eci cally comp osi-
tion, non-colorable knots. We also explore homomorphisms from the knot
group to the dihedral group of a knot and provide computer programs de-
velop ed in our research of knots. We also provide a database and online
resource for future investigations into these arenas.
1 Intro duction
The idea of coloration probably has ro ots as far back as the idea of strands, since
it is but a simple extension of the idea. Since the trefoil knot is tricolorable,
undoubtedly it was this discovery that prompted the study of tricolorability
of other knots and as an invariant. Of course, it was not long after this that
a generalized idea of colorability was conco cted. This general colorability, or
n-Coloration, is the topic of this study.
Since the n-Coloration of a knot can be found using its crossing matrix,
it is relatively easy to calculate. This at rst might seem like a gold mine of
information on all knots, but colorability is not an invariant for all knots, as
we shall see. In fact, its distinguishing p ower is not comparable to anyofthe
other common invariants, but its usefulness lies in b oth its simplicity and its
ease of application. Also, as we shall see, the theory of n-Coloration yields some
interesting relations to existing theory concerning knots.
We rst provide some de nitions to help us maneuver in the language of
knots, colorations, and op erations on these:
De nition 1 A knot is a closed, one dimensional, and non-intersecting curve
in three dimensional space. From a more set-theoretic standpoint, a knot is a
homeomorphism that maps a circle into three dimensional space.
De nition 2 A homeomorphism is general ly consideredtobe an additive and
continuous function. Additive refers to the fol lowing property:
g A + B =g A +g B 1
Also, two objects are considered homeomorphic if there is a homeomorphism
between them.
De nition 3 An ambient isotopy,also known as an isotopy, is a continuous
deformation of a knot or link. This represents the \rubber-sheet geometry" as-
pect of topology, where the knot or link may bebent, twisted, stretched, or pul led.
Under no circumstances, however, may the curve be al lowed to intersect itself
or be cut.
De nition 4 Any group knots or links are considered ambiently isotopic or
isotopic if there exists an ambient isotopy between them. Such a group is cal led
an isotopic class. Al l members of an isotopic class, cal ledprojections, arecon-
sideredtobe the same knot or link.
De nition 5 An isotopic class is a group of knots or links that are ambiently
isotopic, or can be continuously deformed into each other. For example, the
unknot and gure-eight below belong to the same isotopic class because they can
be deformedintoeach other through a typeIReidemeister move.
De nition 6 A pro jection is a speci c member of an isotopic class. For exam-
ple, the unknot and gure-eight below aretwo di erent projections of the same
isotopic class.
De nition 7 The unknot, also known as the \trivial knot", is simply a circle
embeddedinthree-dimensional space with no crossings and al l other projections
of the same isotopic class. The circle and the gure-eight are both considered
De nition 8 A link is a group of knots or unknots embedded in three dimen-
sional space. Each knot or unknot embedded in the link is cal led a component.
A link with only one component is usual ly just referred to as a knot or unknot.
A link in which another member of the link's isotopic class has components
that can each be bounded with non-overlapping spheres, or otherwise separated
components, is cal led the trivial link.
De nition 9 Tricolorability deals with the ability to use three di erent "colors"
to color a knot. A knot is colored by individual strands, where a strand is the
part of a two-dimensional representation of a knot between undercrossings. A
knot is tricolorable if:
Rule 1: At every crossing, either al l threestrands are of a di erent color, or the
same color.
Rule 2: All threecolors areusedincoloring the knot. 2
2 Existence of Colorings
In this section it is shown that the existence of an `-coloration of a given pro jec-
tion of a knot implies the non existence of an `-coloration of any pro jection of
that knot up to Reidemeister moves I, I I, and I I I. Because anyambient isotopy
of a knot compliment space is equivalent to a series of Reidemeister moves and
planar isotopies of a given pro jection of the knot, existence of an `-coloration,
or `-colorability, is a knot invariant.
Given a pro jection of a knot to the plane, lab el its n strands a ;:::;a . Each
1 n
crossing then corresp onds to a linear expression given by: a + a 2a , where
i j k
a and a are the two understrands of the crossing and a is the overcrossing.
i j k
As each strand has exactly two ends, and each crossing has exactly two
understrands or ends, there are always n crossings for given n strands. Thus, a
given knot pro jection and a given lab elling of its n strands determines a so-called
'crossing matrix' of n n dimensions.
Imp ortant: Because a p ermutation of the lab elling corresp onds to a series of
column transp ositions, many imp ortant prop erties i.e. null space, determinant
of the matrix minors up to sign of the matrix are invariant of lab elling.
Tobe`-colorable, a crossing matrix of a pro jection must havea null space
of dimension two or greater when taken mo d `. This means that there is at
least one solution which is fundamentally di erent from the trivial coloration
a;:::;a.
Supp ose a pro jection is `-colorable:
Reidemeister Case 1
In the rst Reidemeister Case, a strand is given a single twist or a single twist
is untwisted. The expression for the crossing of a single twist is a + a 2a .
1 2 1
Whatever ` is, a must b e congruentto a , mo d `. Thus, if a pro jection con-
1 2
tains a single twist and is `-colorable, the two strands must indeed b e considered
as one, 'colorwise', and so a pair of strands which comp ose a single twist is equiv-
alent to a single strand in our coloration system. This shows that colorability
is invariantunderatyp e I Reidemeister move.
Reidemeister Case 2
For the second Reidemeister Case, a lo cal neighb orho o d of the pro jection con-
taining only two disjoint sections of strands which do not cross each other is
transformed such that the piece of the knot corresp onding to a strand a dips
1
underneath the piece of the knot corresp onding to a strand a . There are now
2
four strands a ;a ;a ;a = a see picture and two crossings which b oth
1 2 3 4 1
must satisfy the equation a + a 2a 0 mo d ` if n-colorability is to be
1 3 2
met. Since there is no other restriction on a as a is entirely within the lo cal
3 3
neighb orho o d , there is no p otential change in the rest of the pro jection if we
designate a =2a a . Because nothing is changed outside the lo cal section,
3 2 1 3
colorability is preserved. Conversely, supp ose wehave a lo cal section of a pro jec-
tion whichis`-colorable which contains three sections of strands with numbers
a ;a ;a and a single strand a , and two crossings, a + a 2a 0 mo d `
1 2 3 4 1 4 2
and a + a 2a 0 mo d `. Combining the two equations gives a 2a a
3 4 2 4 2 1
mo d ` a +2a a 2a a a 0 mo d `. Therefore a must be
3 2 1 2 3 1 1
congruentto a for the pro jection to b e `-colorable. If we transform this lo cal
3
section so that a is deleted and the sections with numbers a and a b ecome
4 1 3
one section of strand, the inverse of the Reidemeister move I I, we do not alter
anything outside the lo cal neighb orho o d, and so colorability is preserved under
the second Reidemeister Case.
Reidemeister Case 3
In the Reidemeister Case 3, there is a lo cal section of the pro jection with one
stationary intersection a + a 2a and three strands corresp onding to a section
1 3 2
of the knot passing 'underneath' the lo cal section. If the three strands a ;a ;a
4 5 6
pass on the side of the crossing suchthat a + a 2a 0 mo d ` and a + a
4 5 1 5 6
2a 0 mo d `, then 'shifting' to the other side should not change colorability,
2
0 0
i.e. a + a 2a 0 mo d ` and a + a 2a 0 mo d `. This is b ecause,
4 2 6 3
5 5
if wecombine b oth sets of equations such that the a 's are eliminated, we see
5
that a a + 2a 2a a a + 2a 2a mo d `, which implies
4 6 2 1 4 6 3 2
that a + a 2a 0 mo d `. So the side which a ;a ;a passes under see
1 3 2 4 5 6
picture is trivial i the stationary intersection is consistentwith `-colorability.
Thus colorability is preserved under a Reidemeister moveIII.
3 Op erations on Knots and Their Various Ef-
fects on Coloring
There exist various ways of constructing knots from preexisting ones. \Mu-
tating" a knot and forming satellite knots are two of the simplest. We are
interested in the coloration prop erties mo di ed knots have and their relation to
the coloration prop erties of their \parent" knots.
The easiest way to get a new knot from a pair of known knots is to comp ose
them. This is done by cutting eachofthe \factor" knots op en and attaching
the resulting lo ose strands in suchaway as to create a single knot.
Comp osition of knots can b e generalized in a waythatgives rise to the so-
called satellite knots. Given two knots, K and K , cho ose two tori, T , the
1 2 1
3
b oundary of an neig hbor hood of K and T ,suchthatK S T . Now
1 2 2 2
3
remove the comp onentof S T containing the knot and glue in its place the
1
3
p ortion of S T containing K such that the meridians and longitudes of
2 2
the T are mapp ed, resp ectively, to each other. K and K are known as the
i 1 2
companion and the satellite, resp ectively. A longitude on a torus emb edded in
3
S is simply a curve that forms the b oundary of an orientate surface in the
complementofsaidtorus. 4 A B
C D
Figure 1: A rational tangle.
A B B A
C D D
C
Figure 2: Aknotandamutant. These two are actually equivalent as knots to o!
A satellite of two knots is the same as their sum if the intersection K \ M of
2
K and some meridional disc M of T contains only one p oint. The comp osition
2 2
of K and K will b e denoted K K .
1 2 1 2
Mutating a knot is simpler than the formation of satellite knots. That
simplicity has its price though: if a knot can be p-colored, so can any of its
mutants.
A tangle is like a knot but, instead of being closed, has four points with
neighb orho o ds homeomorphic to [0; 1 , the endp oints of the tangle corresp ond-
3
ing to p oints on the b oundary of a closed ball B S that contains the tangle.
See Figure 1.
Note that a subset of a knot can be a tangle. Simply try to surround a
p ortion of the knot by a sphere such that it is pierced by the knot exactly four
times. The p ortions of the knot \inside" and \outside" of such a sphere are
tangles. Referring to the gure, a mutant of a knot is a new knot obtained by
rotating the sphere, along with it's contents, 180 so that A and B, C and D,
are exchanged. 5 S' KKLL S''
S All one color
Figure 3: Coloring the sum of two knots.
Figure 4: A satellite knot with trefoil as companion and torus knot 2,5 as
satellite
Satellites
One prop erty K L has is that if K is p colorable and L is q colorable, then
K L is b oth p and q colorable. To p color the sum, simply color all of the
strands inherited from K as if coloring K . Supp ose the strand s of K was cut
in the pro cess of forming the sum. If it would have had color c, color the pieces,
0 00
s , s ,ofs, along with all the strands from L, c. See Figure 3. This suggests
the following hyp othesis: If a knot is prime, i.e. isn't the sum of two knots,
then it cannot b e colored in more than one way,formultiple colorings should
corresp ond to factor knots. Alas, this is not true, as will be demonstrated in
Section 4.
How do satellite knots behave under coloration? First, let's lo ok at an
example.
Example 1 Figure 4 shows a satel lite knot with trefoil as companion, and the
2,5 torus knot as satel lite. Watch where those longitudes and meridians go!
A simple but very time consuming, if done by hand! calculation reveals that
this knot is 5 col or abl e, like its satel lite. Note that it isn't 3 col or abl e, like
its companion, the trefoil.
When considering a satellite knot, it may b e of help to consider anyobvious
colorings it may p ossess. Setting K and T as ab ove, it is safe to assume that
i i
K sits in T as in Figure 5. Using this pro jection simpli es the construction
2 2
of the satellite knot. What happ ens when wemakethe satellite knot? If the 6
K
Figure 5: A satellite can sit in its torus like this.
B D A C
A D B
C
Figure 6: Kinoshita-Terasakamutants.
numb er of strands that wrap around T is o dd, we're in business: the satellite
2
knot is colorable as the companion is. The next picture indicates the situation
for satellites with three \strands."The principle generalizes easily. vspace.5cm
R R R B G G G B R R R B
G G G
If the numb er of strands is even we can't sayanything. Figure 4. shows that
satellite knots don't, in general, share colorabilities with their companions.
Mutants
Any mutant ofaknothas the same colorabilities its parent knot has. I hop e
the following example will induce the reader to nd out more.
Example 2 The two knots pictured in Figure 6 have the same colorabilities.
In fact, they can't even be distinguished from the unknot using coloration tech-
niques! 7
4 Multicolorability and Colorability of Higher
Nullities
From section 3 we learned that if the knot K is formed from the comp osition
of the knots K and K which are p and q colorable resp ectively, then K is
1 2
b oth p and q colorable. Thus any comp osite knot is at least bi-colorable if its
comp onents have di erent colorabilities. As we shall see below the converse
is not true. It is natural to ask whether there are any prime knots knots
which are not the sum of two other knots which are multicolorable? If the
comp onents of a knot b oth have the same colorability can anything be said
ab out the comp osite colorability? The de nition of `-colorability is that the
crossing matrix has nullity of at least twomodulo`, are there any knots which
have nullity three or higher? To answer these questions we will rst need to
clarify some de nitions and use our linear algebra skills.
Wesay the knot pro jection K ,with n crossings, is ` colorable if and only if
the crossing matrix C mo d ` has nullity where `; 2 N and 2. Well
K
this is the same thing as saying there exists distinct strands such that: to
each of those strands anyinteger from 0 to ` 1 can b e assigned to the strand
and there exist numb erings colorings of the other n strands which are
dep endent on the numb erings of the strands, but not necessarily uniquely
z x
dep endent such that at eachofthe n crossings: x + y 2 z mo d ` where x; y
tthenumb ers assigned to the understrands and z the numb er assigned
represen y
to the overstrand. See margin.
C is the n n matrix whose rows corresp ond to the equations x + y 2 z =0
K
for the n di erent crossings and the columns are the variables which corresp ond
to the n strands. Let the n 1 column vector v represent a coloration of the knot
th
pro jection, where the i comp onentofv is v and represents the coloring of the
i
th
i strand. In symb olic notation K is ` colorable ,9 2 9 distinct strands
i ;i ; ;i , such that 8 colorings q ;q ; ;q 2 Z , there is a coloration
1 2 1 2 `
n
~v 2 Z with those colorings 8 j 2f1; ; g v = q , such that ~v is a valid
i j
j
`
~
coloration mo dulo ` C ~v 0 mo d ` . Thus there are at least ` di erent
K
`-colorations. For example the simplest case K is `-colorable with nullitytwo
n
~
then 9 i; j j i 6= j 8 q; p 2 Z 9 ~v 2 Z with v = qv = p and C ~v 0 mo d `
` i j K
`
Since there is no privileged frame of reference for the ordering of the strands
in the knot pro jection, the columns of the crossing matrix can be permuted
reordered without a ecting the colorability of the knot. Thus there exits an
n n permutation matrix P such that the distinct strands, whose colorings
can be chosen freely, corresp ond to the last columns of the matrix C P .
K
Using this new permuted crossing matrix we can nd linearly indep endent
coloration vectors v ;v ; ;v of K . These column vectors can be placed
1 2
in a n matrix V , where the columns of V are these coloration vectors.
The span of these vectors is in the mo dular null-space of C P ,however these
K
vectors are not necessarily a basis for the null-space, but if ` is a prime then
they are. Since any integral linear combination of valid colorations is a valid
~
coloration, we can let d be 1 vector that represents a linear combination 8
~
with integer comp onents, then the n 1 column vector V d is a valid coloration
of C P . Knowing that the colorings of the last strands can b e freely chosen
K
and a little linear algebra, we can require the vectors v ;v ; ;v to takeon
1 2
th
the form that the i vector's n i + 1 comp onent is one and all the comp onents
b eneath the one are zero. Thus there is a V of the form
1 0
? ? ? ? ? 1
......
C B
.
......
C B
.
......
C B
C B
? ? ? ? ? 1
C B
C B
1 0 0 0 0 1
C B
. . . .
C B
.
. . . . .
C B
.
. . . 0 1 .
C B
C B
.
.
C B
.
0 0 0 1 0 0
C B
C B
.
.
C B
.
0 0 1 0 0 1
C B
C B
.
.
C B
.
0 1 0 1 0 0
C B
C B
. .
.
. . . A @
.
. . 0 0 1 1
0 0 0 0 0 1
if =3 then V is of the form
0 1
? ? 1
B C
. . .
. . .
B C
. . .
B C
B C
? ? 1
V =
B C
B C
1 0 1
B C
@ A
0 1 1
0 0 1
and the simplest case if =2 then
0 1
? 1
B C
. .
. .
B C
. .
B C
V =
B C
? 1
B C
@ A
1 1
0 1
K is `-colorable with nullity 2 if and only if there exists a p ermu-
tation matrix P and a matrix V of the preceding form such that for all linear
n n n n
~ ~
~
combination vectors d 2 Z the congruence C n P n V d 1 0 1
K 1
`
mo d ` is satis ed.
Ok now that you have explicitly de ned colorability with nullity . How
do es one nd out if a knot is `-colorable, and what value the nullityof C takes
K
on mo dulo `? And once one knows that, how can one nd these vectors whose
span is in the mo dular null-space? Toanswer those questions we simply lo ok
at the Smith and Hermite normal forms of the crossing matrix C . So lets just
K
refresh our memories of what these forms are. 9
The Hermite normal form of a matrix A is an upp er triangular matrix H
with r ank A =thenumb er of nonzero rows of H . The Hermite normal form is
obtained by doing elementary row op erations on A. This includes interchanging
rows, multiplying through a rowby -1, and adding an integral multiple of one
row to another. The Smith normal form S , of an n m rectangular matrix
A of integers, is a diagonal matrix where r ank A = numb er of nonzero rows
r
Q
columns of S ; sig n S =18i; S divides S 8i rank A ; and S
i;i i;i i+1;i+1 i;i
i=1
divides det M for all minors M of rank 0 n Q r ank A =n then jdet A j = S . The Smith normal form is obtained by i;i i=1 doing elementary row and column op erations on A. This includes interchanging rows columns , multiplying through a row column by -1, and adding integral multiples of one row column to another. Since the algorithm for the Hermite normal form do es not include column swapping, all of the colorabilities of K are not necessarily visible along the diagonal of the matrix H . In fact there exist crossing matrixes such that H mo dulo ` only has one row of zeros, despite the fact that H and C b oth have K nullitytwomodulo `. For example if take the following crossing matrix for 7 , 4 we attain the following Hermite form. 1 0 1 0 0 2 0 0 1 C B 1 1 0 0 0 2 0 C B C B 0 1 1 0 2 0 0 C B C B 2 0 1 1 0 0 0 C = K C B C B 0 0 0 1 1 0 2 C B A @ 0 2 0 0 1 1 0 0 0 2 0 0 1 1 0 1 1 0 0 0 2 0 3 B C 0 1 0 0 1 1 3 B C B C 0 0 1 0 0 2 3 B C B C 0 0 0 1 1 0 2 H = B C B C 0 0 0 0 3 3 6 B C @ A 0 0 0 0 0 5 5 0 0 0 0 0 0 0 But if the fth and sixth column of C were to be interchanged then the K diagonal of H would b e the same as the Smith form, namely ve one's, a fteen and a zero. Thus the original Hermite form do esn't explicitly reveal 7 's 15- 4 colorability. Fortunately the Smith normal form do es takeadvantage of column op era- tions so therefor the colorabilities of a knot K along with their corresp onding nullities are readably visible along the diagonal of the Smith normal form S of a crossing matrix C . Namely the knot K is `-colorable if and only if ` divides K the last/greatest non-zero diagonal element of the Smith normal form S . This 10 can be extended to K is `-colorable with nullity if and only if ` divides all comp onents of the last rows and columns of the Smith normal form S . Once the colorabilities of a knot are known the colorations of the pro jection K are easily determined byback substituting the colorings of the freely choiss- able strands into the Hermite normal form H of the crossing matrix C and K solving the corresp onding congruences mo dulo `. Although the strands can b e chosen freely, they do not necessarily de ne a unique coloration. For example the knot 9 has nullity 2 mo d 9 but as the following pictures show, the colorings 48 of the top two strands were xed but the colorings of the lower strands are not uniquely de ned. Well one may think that this implies that the knot 9 has nullity3mod9 48 but this is not the case since these three colorations are the only valid colorations with the top two strands having these values of one and zero. If it had nullity 3 then there would b e nine di erent colorations with the top two strands xed. This phenomena o ccurs when a nontrivial divisor p of ` has WHAT GOES HERE?? Theorem 1 If K is `-colorable with nul lity then 8p j 1 K is p-colorable with nul lity of at least . Pro of 1 After row and column reduction over the integers of the crossing ma- trix C , the Smith normal form has rows which arecongruent to the zerorow K modulo `,ifp divides ` then these same rows are also the zerorow modulo p. ~ ~ If ~x 0 mo d ` and p j ` then ~x 0 mo d p. Theorem 2 If K is p-colorable then for al l k there exist valid non-trivial kp- colorations even if K is not kp-colorable. Pro of 2 Let v be vector corresponding to any valid p-coloration. To obtain a valid kp-coloration, just multiply v by the scalar k and add to that any integral ~ ~ multiple of the vector of al l ones v . If C v 0 mo d p and C v 0 mo d 1 K K 1 ~ kp then 8 mC kv + mv 0 mo d kp. Let u and v be di erent p-colorations K 1 then ku and kv are not necessarily di erent kp-colorations, because ~x 6 ~u mo d p 6, k~x 6 k~u mo d kp Therefor there is no implication that K is kp-colorable. 11 Theorem 3 If K is p-colorable and q -colorable and p and q arerelatively prime then K is also pq -colorable. Pro of 3 Let s be the greatest non-zero diagonal element of the Smith normal form of C then p j s and q j s and gcd p; q =1 pq j s K is pq -colorable. K n 1 Q th th Let L := S 8 i; j jdet M j = L where M is the i j minor of C i;i i;j i;j K i=1 K is not L-colorable ,9` suchthat K is `-colorable with nullity of at least 3. These last theorems put to rest a couple of questions. If a knot is b oth p and q colorable and p 6= q then is the knot a comp osite knot? No. 8 and 9 are b oth 3 and 9 colorable; 7 ; 8 ; 9 ; and 9 are all 3; 5; 20 1 4 21 2 37 and 15 colorable; and 7 ; 8 ; and 9 are all 3; 7; and 21 colorable and all of these 7 5 4 knots are prime. Can you construct a knot that is q -colorable? Yes, if q can b e factored into a nite set of integers which are all relatively prime to each other and eachinteger is less than or equal to 61 then this q -colorable knot can b e constructed from the comp osition of prime knots with nine or less crossings, seeing how 8p 61 there exists a prime knot with crossing numb er less than ten whichisp-colorable. If q can not be factored in such a way then the prime knots of ten and higher crossings have to b e implemented in order to construct this knot. Can you construct a knot that is q -colorable with nullity 3? Yes if q satis es the previous condition then take the knot K whichisq -colorable and comp ose it with itself 1 times. Well those were comp osite knots, are there any prime knots with nullity 3? Yes 8 ; 9 ; 9 ; 9 ; and 9 all havenullity 3 mo dulo 3; 9 ; and 9 18 35 46 47 48 40 49 b oth havenullity3modulo5; and 9 has nullity 3 mo dulo 7. Here are three 41 pictures of the 27 di erent 3-colorations of 9 ,showing how the the colorings of 48 the upp er two strands were xed and the coloring of the lower strand was freely varied, demonstrating how it takes the colorings of three strands to uniquely determine a valid coloration. 12 Figure 7: A f 2; 3; 5g pretzel knot. 5 Visible vs. Invisible Knots Previous sections have b een devoted to nding colorations for knots and de- termining prop erties of those colorations. Here, the goal is to investigate what kinds of knots are colorable and when. Sp eci cally we will explore colorations for pretzel knots and p,p-1 torus knots. Now we will attempt to construct a coloring for the fi,j,kg pretzle knot. For simplicity we will assume that i; j; k > 0, however, none of the following arguments dep end on this and wewillarive at the same conclusions if we allow the signs to vary. De nition 10 Aknot K is cal led invisible if theredoes not exist n such that K is colorable mo d n , and is therefore indistinguishable from the unknot through the scopeofcolorability. Pretzel Knots Recall that a pretzle knot fi; j; k g is constructed from the rational tangles i,j,k by the formula i 0+j 0+ k 0 and then connecting the top two strands together and the b ottom two strands. An example of a f 2; 3; 5g prezle knot is depicted in Fig. 7. In gure 8, wehave colored 5 of the strands a; b; p; q ; r . From the prop erties of colorations as describ ed in sec. Arron or mab ey Allen we know that to any coloration scheme wemay p erform the op erations of adding anyinteger or multiplying byanyinteger and the result will itself b e a coloration. ie, if... 2a b c =0 where a is the over strandandbandc are the understrands 1 then... 2 a + d b + d c + d =0 and... 2af bf cf =0 13 a b x i j k p q r Figure 8: A coloring of a general 3-comp onent pretzel knot. Because of this fact, we can, without loss of generality, let a =0 and b =1, and then use our coloration formula to propagate these colors down the i twists. We quickly see that r = i and p = i +1. Knowing that the incoming strands for the second comp onentof our knot are labled 1 and x, wemay followthe crossing formula down the j twists and nd that p=jx-j+1=i+1 and q= j+1 x-j. Like wise, the third comp onent has incoming strands x and 0 yielding the colors q = 1 k x and r = kx as shown in Figure 9. This gives us three linearly dep endent equations placing restrictions on our variables , two of which are listed b elow. i = kx jx = j + i If wemultiply these equations together our x's fall out and we get the simple equation ji + jk + ik =0 modn 2 and the only time an appropriate n cannot b e chosen to satisfy this equation is when ji + jk + ik = 1 b ecause whenever jji + jk + ik j6=1 we can always cho ose an appropriate n, such that it will equal 0 mo d n. Corolary 1 if i,j, and k share the same sign, then i,j,k is always colorable. pro of: jij + ik + jkj > 1. Corrolary 1 The f-i,2i,2ig pretzel knot is always colorable modanyn. 14 0 1 2 k ij (j+1)x-j=(1-k)x i+1=jx-j+1 i=-kx Figure 9: Some simple relations for the coloring of a general 3 comp onent pretzel knot. PROOF: i 2i + i 2i +2i 2i =0 mo d n 8 n Corollary 1 If the3component pretzel knot fi,j,kg is invisible, then i, j, and k must be relatively prime. Proof: if n devides i and j, then n must devide 1 since i j + i k + j k = 1 Coloring A General Pretzle Knot Wenow generalize this to a k-comp onent pretzle knot K = fC , C ,..., C g. 0 1 k 1 Theorem 1 k 1 X c C C C ::: C ::: C = 1,K invisible 3 0 1 2 i k 1 i=0 Pro of 4 We shal l color the strands of our knot according to the scheme in gure 10, where C 's represent the number of crossings in the j'th tangle component, j a 's are the colors assigned to the strands at the top of those components, and j P 's represent the colors of the strands coming out of the bottom of the pretzle j components. Using the equation for colors at a crossing 2a b c = 0 as b efore we nd that... P = C +1 a C a = C a C 1 a 4 j j 1 j j 1 j 1 j j +1 j j for j 2Z/k Which can b e re-writen as 15 a 0 a k-1 aa1 2 c k-1 c 1 c 0 p k-1 p p 1 2 p 0 Figure 10: A gerneral coloring for a k comp onent pretzel knot. C + C a C a j 1 j j j 1 j 1 a = 5 j +1 C j C + C a C a j 2 j 1 j 1 j 2 j 2 or a = 6 j C j 1 Wesay that a k is the j'th color of a pretzel knot of k comp onents 0...k-1 . j If we let a =0,a = 1 and then apply equation 6we get 0 1 C + C a C a k 1 k i k 1 k 1 7 a i =0= 0 C k It turns out that there is a more simple equation for a k , namely 0 C C C 0 0 0 a k = + + ::: + +1 8 0 C C C 1 2 k 1 which can b e shown by induction and the pro of is omited. >From this we get that K is colorable implies that k 1 X c C C ::: C ::: C 0 mo d n 0 1 i k 1 i=0 K invisible if k 1 X c C C ::: C ::: C = 1 0 1 i k 1 i=0 which ends our pro of. 16 1 i i+1 Figure 11: This showsatypical generator for the braid group on q strands. Figure 12: This shows howtoclosetheendsofabraidknot. Torus Knots De nition 1 ATorus knot T is one that is embedible on the surface of a torus, such that it never intersects itslef. Such a knot can berepresentedbytointegers p,q where p is the number of times the knot loops around the meridian of the torus and q is the number of times it loops around the longitude. Theorem 1 A p,p-1 torus knot is always colorable mo d n . Moreover, if p is odd then n = p or any factor of p and if p is even then n = p 1 or any factor of p-1 . It is easy to show that all torus knots can b e represented by braid knots de- scrib ed in section ?? . They are always of the form p Y ::: = p; q 9 1 2 q 1 i=1 where 's are the q-1 generators see gure 11 of the braid group on q strands. i The knot is then formed by connecting the strands at the top of the braid to those at the b ottom in the cannonical manner depicted in gure 12 In this papp er we will refer to the word ::: as a \gword" and 1 2 q 1 p then represent a torus knot p,q as gword and from nowonwe will only q lo ok at knots of the form p,p-1 . Now let us try to color this p,p-1 torus knot. We start by coloring the strands at the b ottom of our braid knot a ,a ,...,a and then lab eling the 0;1 0;2 0;p 1 17 a p,p-1 ap-1,p-1 a2,p-1 a1,p-1 a 0,1 a0,2 a 0,p-1 Figure 13: This is a general coloration of the p,p-1 torus knot top strands comming out of the i'th gword as a ,a ,..,a however many i;1 i;2 i;p 1 of these colors will b e rep eats, so only the relevent colors are drawn in Figure 13. From the crossing equation we can see that if i p 1 X k p+1 k p+1 [2 1 + 1 k ]a 10 a = i 0;k i;p 1 k =1 where is the kroneker delta function 1 if i = k k = i 0 if i 6= k and if i>1then a = a 11 i;p 1 0;i 1 and for i = p we get p 1 X k p+1 p 1 [2 1 + 1 k ]a 12 a = a = 1 k;p 1 0;p 1 p;p 1 k =1 Let a =0 &a =1 0;1 0;2 and nowtakealookatthecasewhenp isodd. P odd If p is o dd then we can showthat a = i 1. First weknowthatitistruefor 0;i a and a since we colored them 0 and 1 resp ectively. Then assume that it 0;1 o;2 18 is true for a and prove for a . 0;k 0;k +1 So,ifkeven weget a a = k 1 13 0;k 0;1 = 2a0;p 1 2a + ::: a +2a ::: +2a 0;p 2 0;k +1 0;k 0;2 2a0;p 1 2a + ::: 3a +2a ::: +3a 0;p 2 0;k +1 0;k 0;2 = a 1 0;k +1 a = k 0;k +1 andifk isoddweget a a = k 1 14 0;k 0;1 = 2a0;p 1 2a + ::: +3a +2a ::: +2a 0;p 2 0;k +1 0;k 0;2 2a0;p 1 2a + ::: +2a +2a ::: +3a 0;p 2 0;k +1 0;k 0;2 = a 1 0;k +1 and we get the same thing for a so wehaveproven that a must equal 0;k +1 0;k k 1, and our equation for a then simpli es to the following. 0;p 1 a = 0;p 1 p 1 X k 2 1 a a a = k;p 1 1;p 1 p;p 1 k =2 p 1 p 1 X X k k = 2 1 a [2 1 + k ]a 0;k 1 i 0;k 1 k =2 k =1 = 2a 2a +2a + ::: +2a 0;1 0;2 0;3 0;p 2 a +2a 2a + ::: 2a +2a 0;1 0;2 0;3 0;p 2 0;p 1 = 3a 4a +4a ::: +4a0;p 2 2a 0;1 0;2 0;3 0;p 1 Using the fact that a = i 1wenow obtain 0;i p 2 = 3 0 4 1 + 4 2 ::: +4 p 3 2 p 2 4 p 3 3 p 2 = 2 3p 6 = 2p 6 15 p 0 mo d n cho osing n = p or any factor of p will yield a coloration for the knot. Now lets lo ok at the case when p is even. 19 P even Wesaw that when p is o dd wegeta = k 1. If weletp even then we will 0;k see that a k 1 mo d 2 . Weknow that this is true for i =1 and2, so 0;k assume that it is true for k. If k is even then a = 1 and 0;k a a = 1 0;k 0;1 = 2a0;p 1 2a + ::: 3a +2a ::: 2a 0;p 2 0;k +1 0;k 0;2 2a0;p 1 2a + ::: 2a +2a ::: 3a 0;p 2 0;k +1 0;k 0;2 = a +1 0;k +1 a =0 0;k +1 similarly,ifkisoddthen a a = 0 16 0;k 0;1 = 2a0;p 1 2a + ::: + a +2a ::: 2a 0;p 2 0;k +1 0;k 0;2 2a0;p 1 2a + ::: +2a +2a ::: 3a 0;p 2 0;k +1 0;k 0;2 = a +1 0;k +1 a =1 0;k +1 Now pluging this into our equation for a weget i;p 1 a = a =0 17 0;1 2;p 1 p 1 X k 1 = [2 1 k ]a 2 0;k k =1 p 1 X k 1 2 1 a = 2a 3a + 0;k 0;1 0;2 k =3 = 3+ 2a 2a +2a ::: 2a +2a 0;3 0;4 0;5 0;p 2 0;p 1 p 4 = 3 2 2 = p +1 p 1 0 mo d n cho osing n = p 1 orany factor of p-1 will yield a coloration for the knot. Corolary 1 p; q q; p = Corolary 1 A p-1,p torus knot is always colorable mo d n and n = p if p odd, n = p 1 if p even. Corolary 1 A p,p-1 Torus knot with nulity N mo d n can be constructed by letting p betheproduct of N distinct primes. 20 6 n-Coloration as a Prop erty of the Knot Group The fundamental problem of knot theory [1] is the following: Given knots A and B , determine whether they are equivalent under ambient isotopy. The diculty lies in the following: As we compare the prop erties of knot A with the prop er- ties knot B , we might in fact b e comparing prop erties which are not inherent in knottedness itself, but rather in certain representations of knottedness. We might discover something interesting in the pro cess, but we would not be en- gaged in knot theory. The only way to approach this dicultyistousethevery useful idea of an invariant. So, for example, in knot theory we are interested in all prop erties that remain invariantambient isotopy, just as in Euclidean geom- etry we are interested in all prop erties which remain invariant under Euclidean transformation, etc... The rst and rather weak invariantis the notion of n-colorability. At the opp osite end of the scale lies an extremely strong invariant, the fundamental group of the complement of a knot, which we shall from now on refer to as simply the knot group. Since the fundamental group is such a strong invariant, it is natural to ask how it relates to the weakest of invariants, n-colorability. In this section weshow that n-coloration is directly related to a simple prop ertyof the knot group byshowing that a knot is n-colorable if and only if a certain map de nes a homomorphism from the knot group to the dihedral group of order 2n. In order to show this equivalence, we rstgive a brief explanation of the dihedral and knot groups. First, the dihedral group. Consider a regular n-sided p olygon in the Eu- clidian plane. This gure has twoobvious typ es of symmetry: re ections and rotations. These n re ections and n rotations form a group of order 2n denoted by D . Proving these transformations form a group is a simple exercise after n realizing the following: Euclidean transformations are functions and therefore asso ciative under comp osition, a rotation of angle 0 acts as an identity, each re ection is it's own inverse and each rotation has an inverse rotation. This group has the following prop erties: all re ections are of order 2, there exists a minimal rotation of order n which generates all other rotations, and re ec- tions conjugate rotations to their inverses. With this information, wehavethe following presentation of D . n n 2 1 1 Various dihedral groups have as subgroups smaller dihedral groups Ex: Cho os- ing every-other vertex of an o ctagon yields a square: D < D . 4 8 To visualize the knot group, we b egin with an explanation of the fundamental group of a space[3]. Given a top ological space X consider continuous paths in X starting and ending at a certain base point' x 2 X . Notice that we can 0 equate paths that are virtually the same in the following resp ect: eachcanbe continuously 'stretched' into the other without leaving the space. We can also comp ose paths end to end, invert any path, etc... After checking that there is an 'identity' path and that path comp osition is asso ciative, we obtain a group whose elements are the equivalence classes of paths in the space X under what 21 top ologists call homotopy. This group is denoted by X; x , or simply X 0 0 0 if appropriate, and is called the fundamental group of a space X with basep oint x . 0 Such groups are imp ortant in the study of top ological spaces. However, the fundamental group K of a knot itself is rather uninteresting, as the essence 0 of knottedness lies not in the 1-dimensional prop erties of the knot, but in the way this 1-dimensional closed curve is emb edded in 3-dimensions. Therefore we consider instead the fundamental group of the ambient space minus the 3 image of the knot, or S K . Around 1923 a mathematician by the name 0 of Wilhelm Wirtinger discovered a metho d of calculating the knot group of a knot from it's oriented diagram. The metho d is as follows[2]. Given an oriented knot diagram D with strands s ;s ;:::;s 2 Q let T be the set of 1 2 m 3 triples a; b; c 2 Q ,each corresp onding to a particular crossing, where a is the overstrand, b is the incoming understrand, and c is the outgoing understrand. 3 S K isnowgiven by the presentation: 0 1 3 1 2 m Verbally, this could b e stated as follows: the overstrand conjugates the in- coming understrand to the outgoing understrand, with each strand asso ciated to the path that leaves the base p oint, lo ops around the oriented strand according to the right hand rule, and returns to the base p oint. It is a rather interesting 1 visual exercise to show that the 'Wirtinger Relation,' namely a ba = c, holds. 3 We now attempt to de ne a homomorphism from S K to D by 0 n 3 de ning it's action on the generators of S K . Using the conjugacy relation 0 k i in the dihedral presentation as a hint, wetry : s ! yx where k : Q ! Z. i This leads to the following: Theorem 4 Let D be an oriented knot diagram, and let K be the knot it repre- sents. D is n-colorable if and only if there exists a homomorphism from the 3 k i knot group S K to D where sends each s to yx 0 n i Pro of 5 Tocheck the validity of the homomorphism, we determine under which 3 conditions the image of S K under satis es the Wirtinger relation for 0 al l a; b; c 2 T , which leads us to the fol lowing equivalent equations: 1 a ba = c k a 1 k b k a k c yx yx yx = yx k a 1 k b k a k c x y yx yx = yx k b k a k a k c x yx = yx k b k a k a k c x yx y = yx y k b 2k a k c x = x 2k a k b k c = 0modn 22 Notice the last equation is, with our notation, the condition for the n-colorability ofaknot,withthecoloration function v q =k q modn. We have now shown that the concept of n-colorability is encapsulated in the fundamental group of a knot. Couldn't analogous results hold for other knot-theoretic concepts? For instance, supp ose for some strange reason history to ok a di erent course and we had the concepts of the knot p olynomial and the knot group but not the concept of n-colorability. Then, while studying the knot group, we ask "With which knots, if any, do es there exist a homomorphism to a dihedral group?" After a little exp erimentation, we obtain the homomorphism de ned by , and derive the concept of colorability from group theoretic prop- erties of the knot group. Likewise, wecannow ask ourselves, given a class of groups G, "With which knots, if any, do es there exist a homomorphism from the knot group to a group in G?" By determining such conditions, wecanper- haps derive new knots-theoretic concepts from group theoretic prop erties. The rst natural question to ask is: Which concepts corresp ond to the existence of homomorphisms from the knot group to the class of symmetric groups? 7 Summary of work This pap er has discussed the idea of a \coloration" of a knot whichshows that a knot is di erent from the unknot. Wehaveshown that an example of this is \tri-colorability" which extends naturally to \n-colorability." Wehaveshown the existence of n-colorations of knots through computation by insp ection. Also wehave used the crossing matrix to compute the number of colors needed to compute the coloration of a given knot. This results in a matrix with a nullityoftwo mo dulo some integer whichisthenumb er of colors. There are prime knots and comp osite knots. Like the integers, a knot is either comp osite or prime. And liketheintegers a comp osite knot is the unique comp osition of primes. We have observed that some crossing matrices has a nullity greater than two. Many of these knots are pretzel knots. Unfortunately n-colorabilityof knots is not a universal detector of the un- knot. There are some knots which have a Jones p olynomial which is di er- ent from the unknot, which are not colorable. Hence in these situations n- colorability fails as a test for the unknot. Wehave explored representations of the fundamental group into the dihedral group. Finally wehave computed the crossing matrices for all of the knots app earing in our text. These were included in the preceding section \Tables of matrices." This demonstrates the wide range of usabilityofthistechnique. 23 8 Conclusions This pap er represents a sample of the work done in eight short weeks under- standing the theory of knots. Additional time was sp entinanoverview of groups and hyp erb olic geometry and the application of each to the theory of knots. All were given opp ortunitytoteach and share the things that they had learned. References [1] C.C. Adams, The Knot Book. W.H. Freeman and Company: New York, 2001. [2] W.S. Massey, A Basic Cource In Algebraic Topology, Graduate Texts in Mathematics, vol. 127. Springer-Verlag: New York, 1991. [3] J.R. Munkres, Topology, 2nd Ed. Prentice-Hall: New Jersey,1999. [4] K. Reidemeister, Knotentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, Alte Folge , Band 1, Heft 1. 24 A Tables of Matrices The following crossing matrices are in 1-1 corresp ondence with the table of knots in the app endix of ?? . Someb o dy please tell me how to get the two-column stu to end up on this page to o! 25 7-4 3-1 0 1 1 2 1 @ A 1 1 2 0 1 1 0 0 2 0 0 1 2 1 1 1 1 0 0 2 0 0 B C B C 0 1 1 0 0 2 0 B C 4-1 B C 0 1 2 0 1 1 0 0 0 B C 1 0 2 1 B C 0 0 0 1 1 0 2 B C B C 1 1 0 2 B C @ A 0 0 2 0 1 1 0 @ A 2 1 1 0 0 2 0 0 0 1 1 0 2 1 1 5-1 0 1 7-5 1 0 2 0 1 B C 1 1 0 2 0 B C 0 1 B C 0 1 1 0 2 1 0 2 0 0 0 1 B C @ A 2 0 1 1 0 1 1 0 0 2 0 0 B C B C 0 2 0 1 1 0 1 1 0 0 0 2 B C B C 0 0 1 1 0 2 0 B C B C 5-2 2 0 0 1 1 0 0 B C 0 1 @ A 1 0 2 0 1 0 0 0 2 1 1 0 B C 1 1 0 2 0 0 2 0 0 0 1 1 B C B C 2 1 1 0 0 B C @ A 0 0 1 1 2 7-6 0 2 0 1 1 0 1 6-1 1 2 0 0 0 0 1 1 1 0 0 2 0 0 B C B C 0 1 0 1 1 0 0 0 2 B C 1 0 2 0 0 1 B C 0 0 1 1 0 2 0 B C 1 1 0 0 2 0 B C B C B C 0 0 2 1 1 0 0 B C 2 1 1 0 0 0 B C @ A B C 2 0 0 0 1 1 0 0 0 1 1 0 2 B C @ A 0 0 0 2 0 1 1 0 2 0 1 1 0 0 0 0 2 1 1 7-7 6-2 0 1 1 0 0 2 0 0 1 1 0 1 1 2 0 0 0 0 B C 1 0 2 0 0 1 B C 0 1 1 0 0 2 0 B C 1 1 0 2 0 0 C B B C C B 0 0 1 1 2 0 0 B C 0 1 1 0 0 2 C B B C C B 0 0 0 1 1 0 2 B C 0 0 1 1 2 0 C B @ A A @ 2 0 0 0 1 1 0 2 0 0 1 1 0 0 2 0 0 0 1 1 0 2 0 0 1 1 8-1 6-3 0 1 0 1 1 0 2 0 0 0 0 1 1 0 0 2 0 1 1 1 0 0 0 2 0 0 B C 1 1 2 0 0 0 B C B C B C 2 1 1 0 0 0 0 0 B C 0 1 1 0 0 2 B C B C B C 0 0 1 1 0 0 0 2 B C 0 0 1 1 2 0 B C B C @ A 0 0 0 1 1 0 2 0 B C 2 0 0 1 1 0 B C 0 2 0 0 1 1 0 0 B C 0 2 0 0 1 1 @ A 0 0 0 0 2 1 1 0 0 0 0 2 0 0 1 1 7-1 8-2 0 1 1 0 0 2 0 0 1 1 1 0 0 2 0 0 B C 1 0 B C 1 0 0 2 0 0 0 1 0 1 1 0 0 2 0 B C B C 1 1 0 0 2 0 0 0 C B 0 0 1 1 0 0 2 B C C B B C 0 1 1 0 0 0 2 0 C B 2 0 0 1 1 0 0 B C C B @ A 0 0 1 1 0 0 0 2 C B 0 2 0 0 1 1 0 C B 0 0 0 1 1 2 0 0 C B 0 0 2 0 0 1 1 C B 2 0 0 0 1 1 0 0 C B A @ 0 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 7-2 1 0 1 0 2 0 0 0 1 8-3 1 1 0 0 2 0 0 C B C B 2 1 1 0 0 0 0 C B 1 0 1 0 0 2 0 0 0 1 C B 0 0 1 1 0 0 2 C B 1 1 0 0 0 0 2 0 C B C B 0 0 0 1 1 2 0 C B C B 0 1 1 0 0 2 0 0 A @ C B 0 2 0 0 1 1 0 C B 2 0 1 1 0 0 0 0 C B 0 0 0 2 0 1 1 C B 0 0 0 1 1 0 0 2 C B C B 0 0 2 0 1 1 0 0 C B A @ 0 2 0 0 0 1 1 0 7-3 0 0 0 0 2 0 1 1 0 1 1 0 0 2 0 0 1 8-4 1 1 0 0 0 2 0 B C B C 0 1 1 0 2 0 0 B C B C 0 0 1 1 0 0 2 B C 1 0 1 0 0 0 2 0 0 1 B C 2 0 0 1 1 0 0 B C 1 1 0 2 0 0 0 0 C B @ A 0 2 0 0 1 1 0 C B 0 1 1 0 0 0 2 0 C B 0 0 2 0 0 1 1 C B 0 0 1 1 0 2 0 0 C B C B 0 0 0 1 1 0 0 2 C B C B 2 0 0 0 1 1 0 0 C B A @ 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 26 8-13 8-5 0 1 0 1 1 0 2 0 0 0 0 1 1 0 2 0 0 0 0 1 1 1 0 0 0 2 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 0 1 1 0 0 0 2 0 0 1 1 0 0 0 0 2 B C B C B C B C 0 0 1 1 0 0 0 2 2 0 1 1 0 0 0 0 B C B C B C B C 0 2 0 1 1 0 0 0 0 0 0 1 1 0 2 0 B C B C B C B C 2 0 0 0 1 1 0 0 0 2 0 0 1 1 0 0 B C B C @ A @ A 0 0 0 0 2 1 1 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 2 0 1 1 8-14 8-6 0 1 0 1 1 0 0 2 0 0 0 1 1 0 2 0 0 0 0 1 1 1 0 0 0 2 0 0 1 1 0 0 2 0 0 0 B C B C B C B C 0 1 1 0 0 0 0 2 0 1 1 0 0 0 0 2 B C B C B C B C 2 0 1 1 0 0 0 0 0 0 1 1 0 0 2 0 B C B C B C B C 0 0 2 1 1 0 0 0 0 0 0 1 1 2 0 0 B C B C B C B C 0 0 0 0 1 1 2 0 2 0 0 0 1 1 0 0 B C B C @ A @ A 0 2 0 0 0 1 1 0 0 0 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 2 0 0 0 0 1 1 8-15 8-7 0 1 0 1 1 0 2 0 0 0 0 1 1 0 0 0 0 2 0 1 1 1 0 0 0 2 0 0 1 1 0 0 2 0 0 0 B C B C B C B C 0 1 1 0 0 0 0 2 2 1 1 0 0 0 0 0 B C B C B C B C 0 2 1 1 0 0 0 0 0 0 1 1 0 0 2 0 B C B C B C B C 0 0 0 1 1 0 2 0 0 0 0 1 1 0 0 2 B C B C B C B C 0 0 0 2 1 1 0 0 0 2 0 0 1 1 0 0 B C B C @ A @ A 2 0 0 0 0 1 1 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 2 0 0 1 1 8-16 8-8 1 0 1 0 1 0 2 0 0 0 0 1 1 0 0 0 0 0 2 1 1 1 0 2 0 0 0 0 1 1 0 0 0 2 0 0 C B C B C B C B 0 1 1 0 0 0 0 2 0 1 1 0 2 0 0 0 C B C B C B C B 0 0 1 1 0 2 0 0 0 0 1 1 0 0 0 2 C B C B C B C B 0 0 0 1 1 0 2 0 0 2 0 1 1 0 0 0 C B C B C B C B 0 2 0 0 1 1 0 0 0 0 0 2 1 1 0 0 C B C B A @ A @ 2 0 0 0 0 1 1 0 2 0 0 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 2 0 0 0 1 1 8-17 8-9 0 1 0 1 1 0 0 0 0 2 0 1 1 0 0 2 0 0 0 1 1 1 0 0 0 0 2 0 1 1 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 2 0 0 0 0 1 1 0 0 2 0 0 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 1 1 0 0 0 2 B C B C B C B C 2 0 0 1 1 0 0 0 2 0 0 1 1 0 0 0 B C B C B C B C 0 0 2 0 1 1 0 0 0 2 0 0 1 1 0 0 B C B C @ A @ A 0 0 0 2 0 1 1 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 2 0 1 1 8-18 8-10 1 0 1 0 1 0 0 0 0 2 0 1 1 0 2 0 0 0 0 1 1 1 0 0 0 0 2 0 1 1 0 0 2 0 0 0 C B C B C B C B 0 1 1 0 0 0 0 2 0 1 1 0 0 0 0 2 C B C B C B C B 2 0 1 1 0 0 0 0 2 0 1 1 0 0 0 0 C B C B C B C B 0 2 0 1 1 0 0 0 0 0 0 1 1 0 2 0 C B C B C B C B 0 0 2 0 1 1 0 0 0 2 0 0 1 1 0 0 C B C B A @ A @ 0 0 0 2 0 1 1 0 0 0 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 0 2 1 1 8-19 8-11 0 1 0 1 1 0 2 0 0 0 0 1 1 0 0 2 0 0 0 1 1 1 0 0 2 0 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 0 1 1 0 0 0 0 2 0 1 1 0 0 0 2 0 B C B C B C B C 0 0 1 1 0 2 0 0 0 0 1 1 2 0 0 0 B C B C B C B C 2 0 0 1 1 0 0 0 0 0 0 1 1 0 0 2 B C B C B C B C 0 0 2 0 1 1 0 0 0 2 0 0 1 1 0 0 B C B C @ A @ A 2 0 0 0 0 1 1 0 2 0 0 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 2 0 0 0 1 1 8-20 8-12 1 1 0 0 1 0 2 0 0 0 0 1 1 0 0 2 0 0 0 1 1 1 0 0 0 0 2 0 1 1 0 0 0 0 2 0 C C B B C C B B 0 1 1 0 2 0 0 0 0 1 1 0 2 0 0 0 C C B B C C B B 0 0 1 1 0 0 2 0 2 0 1 1 0 0 0 0 C C B B C C B B 2 0 0 1 1 0 0 0 0 0 2 1 1 0 0 0 C C B B C C B B 0 0 2 0 1 1 0 0 0 0 0 0 1 1 0 2 C C B B A A @ @ 2 0 0 0 0 1 1 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 0 2 1 1 27 9-7 8-21 1 0 1 0 1 0 2 0 0 0 0 0 1 1 0 0 0 0 0 2 1 1 1 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 C B C B C B C B 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 2 C B C B C B C B 0 0 1 1 0 0 0 2 0 0 0 1 1 0 2 0 0 C B C B C B C B 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 0 2 C B C B C B C B 2 0 0 0 1 1 0 0 0 0 0 0 0 1 1 2 0 C B C B C B A @ 0 0 0 0 2 1 1 0 0 0 0 0 2 0 1 1 0 C B A @ 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 9-1 9-8 1 0 1 0 0 0 2 0 0 0 1 1 0 1 2 0 0 0 0 0 0 1 1 1 0 0 0 2 0 0 0 C B C B 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 C B C B C B C B 0 1 1 0 0 0 0 0 2 0 0 1 1 0 0 0 2 0 C B C B C B C B 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 2 C B C B C B C B 0 0 0 1 1 0 2 0 0 2 0 0 0 1 1 0 0 0 C B C B C B C B 0 0 0 2 1 1 0 0 0 0 2 0 0 0 1 1 0 0 C B C B C B A @ 2 0 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 C B A @ 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 1 1 9-2 9-9 0 1 1 0 2 0 0 0 0 0 1 0 1 1 0 0 2 0 0 0 0 1 1 1 0 0 0 2 0 0 0 B C B C 1 1 0 0 2 0 0 0 0 2 1 1 0 0 0 0 0 0 B C B C B C B C 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 0 2 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 2 0 B C B C B C B C 2 0 0 1 1 0 0 0 0 0 0 0 0 1 1 2 0 0 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 2 0 0 0 1 1 0 0 B C B C B C @ A 0 2 0 0 0 1 1 0 0 0 0 0 0 2 0 1 1 0 B C @ A 0 0 2 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 0 0 2 0 1 1 9-3 9-10 0 1 1 0 0 0 2 0 0 0 1 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 0 2 0 0 B C B C 1 1 0 0 2 0 0 0 0 0 1 1 0 0 2 0 0 0 B C B C B C B C 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 2 0 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 2 B C B C B C B C 0 0 0 1 1 0 0 0 2 2 0 0 0 1 1 0 0 0 B C B C B C B C 2 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 B C B C B C @ A 0 0 2 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 B C @ A 0 2 0 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 9-4 9-11 0 1 1 0 0 2 0 0 0 0 1 0 1 1 0 2 0 0 0 0 0 1 1 1 0 0 0 2 0 0 0 B C B C 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 2 1 1 0 0 0 0 0 0 2 0 1 1 0 0 0 0 0 B C B C B C B C 0 0 1 1 0 2 0 0 0 0 0 0 1 1 0 0 0 2 B C B C B C B C 0 0 0 1 1 0 0 2 0 0 0 0 0 1 1 0 2 0 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 2 0 0 0 1 1 0 0 B C B C B C @ A 0 2 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 B C @ A 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 0 1 1 0 0 0 0 2 0 0 1 1 9-5 9-12 0 1 1 0 0 0 2 0 0 0 1 0 1 1 0 2 0 0 0 0 0 1 1 1 0 0 0 2 0 0 0 B C B C 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 0 2 0 0 0 0 0 1 1 0 0 2 0 0 B C B C B C B C 0 0 1 1 0 0 0 0 2 2 0 0 1 1 0 0 0 0 B C B C B C B C 0 0 0 1 1 0 0 2 0 0 0 0 0 1 1 0 0 2 B C B C B C B C 0 0 0 2 1 1 0 0 0 0 0 0 2 0 1 1 0 0 B C B C B C @ A 0 2 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 B C @ A 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 0 1 1 0 0 0 0 2 0 0 1 1 9-6 9-13 0 1 1 0 0 2 0 0 0 0 1 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 2 0 0 0 B C B C 1 1 0 0 2 0 0 0 0 0 1 1 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 0 0 2 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 2 0 0 B C B C B C B C 2 0 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 B C B C B C B C 0 2 0 0 1 1 0 0 0 0 0 0 0 2 1 1 0 0 B C B C B C @ A 0 0 0 0 0 1 1 0 2 0 2 0 0 0 0 1 1 0 B C @ A 0 0 0 2 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 28 9-21 9-14 1 0 1 0 1 0 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 0 2 0 0 0 0 C B C B C B C B 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 0 2 0 C B C B C B C B 0 0 1 1 0 2 0 0 0 0 0 1 1 0 0 2 0 0 C B C B C B C B 0 0 0 1 1 0 0 0 2 0 0 0 1 1 2 0 0 0 C B C B C B C B 0 0 0 2 1 1 0 0 0 2 0 0 0 1 1 0 0 0 C B C B C B C B 2 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 C B C B A @ A @ 0 2 0 0 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 0 0 2 0 0 1 1 0 2 0 0 0 0 0 1 1 9-22 9-15 1 0 1 0 1 0 0 0 2 0 0 0 1 1 0 2 0 0 0 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 0 0 2 0 0 0 C B C B C B C B 0 1 1 0 0 0 0 0 2 0 1 1 2 0 0 0 0 0 C B C B C B C B 0 0 1 1 0 2 0 0 0 0 0 1 1 0 0 0 0 2 C B C B C B C B 2 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 2 0 C B C B C B C B 0 0 2 0 1 1 0 0 0 0 0 0 0 1 1 2 0 0 C B C B C B C B 0 0 0 2 0 1 1 0 0 0 2 0 0 0 1 1 0 0 C B C B A @ A @ 0 2 0 0 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 0 0 0 2 1 1 2 0 0 0 0 0 0 1 1 9-23 9-16 0 1 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 2 0 0 0 0 0 B C B C B C B C 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 0 0 1 1 0 0 0 0 2 2 0 1 1 0 0 0 0 0 B C B C B C B C 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 B C B C B C B C 0 0 0 0 1 1 2 0 0 0 0 0 0 1 1 0 2 0 B C B C B C B C 0 0 2 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 B C B C @ A @ A 0 0 0 0 2 0 1 1 0 0 0 2 0 0 0 1 1 0 2 0 0 0 0 0 0 1 1 0 0 0 0 0 2 0 1 1 9-24 9-17 0 1 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 B C B C B C B C 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 2 0 B C B C B C B C 0 0 1 1 2 0 0 0 0 0 0 1 1 0 0 0 0 2 B C B C B C B C 0 0 0 1 1 0 0 0 2 2 0 0 1 1 0 0 0 0 B C B C B C B C 0 0 0 0 1 1 2 0 0 0 0 0 2 1 1 0 0 0 B C B C B C B C 0 0 2 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 B C B C @ A @ A 2 0 0 0 0 0 1 1 0 0 0 0 0 0 2 1 1 0 0 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 9-25 9-18 0 1 0 1 1 0 2 0 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 1 0 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 B C B C B C B C 0 1 1 0 2 0 0 0 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 2 B C B C B C B C 0 2 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 B C B C B C B C 0 0 0 2 1 1 0 0 0 0 0 0 0 1 1 0 2 0 B C B C B C B C 0 0 0 0 0 1 1 2 0 0 2 0 0 0 1 1 0 0 B C B C @ A @ A 2 0 0 0 0 0 1 1 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 2 0 1 1 0 0 0 2 0 0 0 1 1 9-26 9-19 0 1 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 0 2 0 1 1 1 2 0 0 0 0 0 0 1 1 0 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 0 2 0 0 0 2 1 1 0 0 0 0 0 0 B C B C B C B C 0 0 1 1 0 0 2 0 0 0 0 1 1 0 2 0 0 0 B C B C B C B C 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 0 0 2 1 1 0 0 0 B C B C B C B C 0 2 0 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 B C B C @ A @ A 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 0 2 0 0 1 1 0 0 0 0 2 0 0 1 1 9-27 9-20 0 1 0 1 1 2 0 0 0 0 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 B C B C B C B C 0 1 1 0 0 0 0 2 0 2 1 1 0 0 0 0 0 0 B C B C B C B C 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 2 0 0 B C B C B C B C 0 0 2 1 1 0 0 0 0 0 0 0 1 1 0 0 2 0 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 2 0 0 1 1 0 0 0 B C B C B C B C 2 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 B C B C @ A @ A 0 0 0 2 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 29 9-35 9-28 1 0 1 0 1 0 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 1 1 1 0 0 2 0 0 0 0 1 1 2 0 0 0 0 0 0 C B C B C B C B 0 1 1 0 0 0 2 0 0 0 1 1 0 2 0 0 0 0 C B C B C B C B 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 2 0 C B C B C B C B 0 0 0 1 1 0 0 2 0 0 2 0 1 1 0 0 0 0 C B C B C B C B 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 C B C B C B C B 0 0 2 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 C B C B A @ A @ 0 2 0 0 0 0 1 1 0 0 0 0 0 0 2 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 9-36 9-29 1 0 1 0 1 0 2 0 0 0 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 2 0 0 0 0 1 1 0 2 0 0 0 0 0 C B C B C B C B 2 1 1 0 0 0 0 0 0 0 1 1 0 0 0 2 0 0 C B C B C B C B 0 0 1 1 0 0 2 0 0 2 0 1 1 0 0 0 0 0 C B C B C B C B 0 2 0 1 1 0 0 0 0 0 0 0 1 1 0 0 2 0 C B C B C B C B 0 0 0 0 1 1 0 2 0 0 0 2 0 1 1 0 0 0 C B C B C B C B 0 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 2 C B C B A @ A @ 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 0 0 2 0 1 1 0 2 0 0 0 0 0 1 1 9-37 9-30 0 1 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 B C B C B C B C 0 1 1 2 0 0 0 0 0 2 1 1 0 0 0 0 0 0 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 1 1 2 0 0 0 0 B C B C B C B C 0 0 0 1 1 2 0 0 0 0 0 0 1 1 0 2 0 0 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 2 B C B C B C B C 0 2 0 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 B C B C @ A @ A 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 9-38 9-31 0 1 0 1 1 0 2 0 0 0 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 2 0 0 0 1 1 0 0 0 0 2 0 0 B C B C B C B C 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 2 0 B C B C B C B C 0 0 1 1 0 0 2 0 0 0 2 1 1 0 0 0 0 0 B C B C B C B C 2 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 B C B C B C B C 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 0 2 B C B C B C B C 0 0 0 2 0 1 1 0 0 0 0 0 0 2 1 1 0 0 B C B C @ A @ A 0 2 0 0 0 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 0 1 1 0 0 2 0 0 0 0 1 1 9-39 9-32 0 1 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 0 2 0 0 1 1 0 2 0 0 0 0 0 B C B C B C B C 0 1 1 0 0 2 0 0 0 0 1 1 0 2 0 0 0 0 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 0 0 B C B C B C B C 0 2 0 1 1 0 0 0 0 0 0 0 1 1 0 0 2 0 B C B C B C B C 0 0 0 0 1 1 0 0 2 0 0 2 0 1 1 0 0 0 B C B C B C B C 0 0 0 2 0 1 1 0 0 0 0 0 0 0 1 1 0 2 B C B C @ A @ A 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 2 0 0 0 0 1 1 2 0 0 0 0 0 0 1 1 9-40 9-33 0 1 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 0 0 0 0 1 1 1 0 2 0 0 0 0 0 1 1 0 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 0 0 0 0 2 2 1 1 0 0 0 0 0 0 B C B C B C B C 0 0 1 1 0 0 0 2 0 0 0 1 1 0 2 0 0 0 B C B C B C B C 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 2 0 0 B C B C B C B C 0 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 B C B C B C B C 0 2 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 B C B C @ A @ A 2 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 1 1 9-41 9-34 0 1 0 1 1 0 0 0 0 2 0 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 0 2 0 B C B C B C B C 0 1 1 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 B C B C B C B C 0 0 1 1 0 0 0 0 2 2 0 1 1 0 0 0 0 0 B C B C B C B C 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 B C B C B C B C 2 0 0 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 B C B C B C B C 0 0 2 0 0 1 1 0 0 0 0 0 0 0 1 1 0 2 B C B C @ A @ A 0 0 0 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 30 9-49 9-42 1 0 1 0 1 0 0 0 2 0 0 0 1 1 0 0 0 2 0 0 0 1 1 1 0 0 0 0 0 2 0 1 1 0 0 0 0 0 2 0 C B C B C B C B 0 1 1 0 0 2 0 0 0 0 1 1 0 2 0 0 0 0 C B C B C B C B 0 0 1 1 0 0 0 2 0 0 0 1 1 0 0 2 0 0 C B C B C B C B 0 2 0 1 1 0 0 0 0 0 0 2 1 1 0 0 0 0 C B C B C B C B 0 0 0 0 1 1 0 0 2 2 0 0 0 1 1 0 0 0 C B C B C B C B 0 0 0 2 0 1 1 0 0 0 0 0 2 0 1 1 0 0 C B C B A @ A @ 0 2 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 2 0 0 0 0 0 1 1 9-43 1 0 1 0 0 2 0 0 0 0 1 1 1 0 0 0 0 2 0 0 C B C B 0 1 1 0 0 0 0 0 2 C B C B 0 2 1 1 0 0 0 0 0 C B C B 0 0 0 1 1 0 2 0 0 C B C B 0 0 0 2 1 1 0 0 0 C B C B 2 0 0 0 0 1 1 0 0 C B A @ 0 0 0 0 2 0 1 1 0 0 0 2 0 0 0 0 1 1 9-44 0 1 1 0 2 0 0 0 0 0 1 1 1 0 0 0 0 0 2 0 B C B C 0 1 1 0 0 0 0 0 2 B C B C 0 2 1 1 0 0 0 0 0 B C B C 0 0 0 1 1 2 0 0 0 B C B C 0 0 0 0 1 1 0 2 0 B C B C 0 2 0 0 0 1 1 0 0 B C @ A 0 0 2 0 0 0 1 1 0 0 0 0 0 2 0 0 1 1 9-45 0 1 1 0 0 0 0 0 2 0 1 1 1 0 0 2 0 0 0 0 B C B C 0 1 1 0 0 2 0 0 0 B C B C 0 0 1 1 0 0 2 0 0 B C B C 0 2 0 1 1 0 0 0 0 B C B C 0 0 0 2 1 1 0 0 0 B C B C 0 0 0 0 0 1 1 0 2 B C @ A 0 0 2 0 0 0 1 1 0 0 0 0 0 0 2 0 1 1 9-46 0 1 1 0 0 0 2 0 0 0 1 1 1 0 2 0 0 0 0 0 B C B C 0 1 1 0 0 2 0 0 0 B C B C 0 0 1 1 0 0 0 2 0 B C B C 2 0 0 1 1 0 0 0 0 B C B C 0 0 2 0 1 1 0 0 0 B C B C 0 2 0 0 0 1 1 0 0 B C @ A 0 0 0 2 0 0 1 1 0 0 0 2 0 0 0 0 1 1 9-47 0 1 1 0 0 0 0 0 2 0 1 1 1 0 0 0 0 0 2 0 B C B C 0 1 1 0 0 0 2 0 0 B C B C 2 0 1 1 0 0 0 0 0 B C B C 0 2 0 1 1 0 0 0 0 B C B C 0 0 2 0 1 1 0 0 0 B C B C 2 0 0 0 0 1 1 0 0 B C @ A 0 0 0 0 2 0 1 1 0 0 0 0 2 0 0 0 1 1 9-48 0 1 1 0 0 0 0 2 0 0 1 1 1 0 0 0 0 0 2 0 B C B C 0 1 1 0 0 2 0 0 0 B C B C 0 0 1 1 0 0 2 0 0 B C B C 0 0 0 1 1 0 0 2 0 B C B C 0 0 0 0 1 1 2 0 0 B C B C 0 0 0 0 0 1 1 0 2 B C @ A 0 2 0 0 0 0 1 1 0 0 0 0 2 0 0 0 1 1 31