Bu makale 8 10 düğümü ve onun örgüsü ile ilgilidir. Düğüm teorisinde önemli kavramlardan birisi olan örgü kavramını kullanarak 8 10 düğümünün örgüsünün Artin ve Garside temsilini elde edeceğiz. Daha sonra pozitiflik kavramı irdelenecektir.This paper is concerned with 810 knots and its braids. The braids structure is very important role in Knots Theory. In view of this structure, we obtained braids for that knot and we will give the representations of Artin and we examine Garside Word problem. And then we will examine the positivity structure for these knots

KÜÇÜK, Ahmet

ŞİMŞEK, Hakan

English

DoÄuÅ Ãniversitesi Dergisi

Doğuş Üniversitesi Dergisi, 2001 / 4
ON THE BRAIDS FOR 810 KNOT
Hakan ŞİMŞEK
Atatürk Üniversitesi, Fen Edebiyat Fakültesi, M atematik Bölümü
Ahmet KÜÇÜK
Atatürk Üniversitesi, Fen Edebiyat Fakültesi, M atematik Bölümü
SUMMARY: This paper is concerned with 8 1 0  knots and its braids. The braids 
structure plays a very important role in Knots Theory. In view of this structure, we 
obtain braids for that knot, give the representations of Artin and examine Garside 
Word problem. Then we examine the positivity structure for these knots.
Key words: Braids, Positive word, Representations of Braids.
ÖZET: Bu makale 8 jo düğümü ve onun örgüsü ile ilgilidir. Düğüm teorisinde 
önemli kavramlardan birisi olan örgü kavramını kullanarak 8 jo düğümünün örgüsü­
nün Artin ve Garside temsilini elde edeceğiz. Daha sonra pozitiflik kavramı irdele­
necektir.
Anahtar Kelimeler: Örgü, Pozitif kelime, Örgü Temsili.
INTRODUCTION: The word problem in Bn was solved by Artin in (Artin, E., 
1925). His solution was based on his knowledge of structure of the kernel of the map 
(j> from from B n to the symétrie group Xn which sends the generator 5j to the 
transposition (/,/+1). The Conjugacy problem in Bn was also posed in (Artin, E., 
1925), also its importance for the problem of recognizing knots and links 
algorithmicaly was noted, however it took 43 years before progress was made. In a 
different, but equally foundational manuscript (Garside, F.A., 1969), he discovered 
a new solution to the word problem which then led him to a related solution to the 
conjugacy problem (and also [Birman, J., 1998]).
A somewhat different question is the shortest word problem, to find a representative 
of the word class which has shortest length in the Artin generators. It was proved in 
(Paterson, M.S., 1991) that this problem in Bn is at last as hard as an NP-complete 
problem. Garside and Thurston and Birman, using new generators, will be able to 
solve the word problem.
We gave the Braid which is related to Artin and Garside generators and investigated 
positivity structure for 8 1 0  knot.
An n-braid is a very particular example of an (n)- tangle. On the top and base of a
Hakan Şimşek, Ahmet Küçük 106
These points may be arbitrarily placed, however we shall express them in terms of 
specific coordinates.
Firstly, the coordinates for B in R3 are,
B = { ( x , y , z ) \ 0 <x , y , z < \ }
Let us choose A l , A 2, • • •, An , A [, A '2, • • •, A'n as follows,
2 n + 1 2 n + 1
2 n +1 2 « + l
By the construction each A' is directly below the corresponding A, Figure 1. Now 
join the A x, A 2, • • •, An , to A[, A'2 A'n by means of n curves in B. As usual,
they are joined in such a way that these curves (including the end points) do not 
mutually intersect each other. We will call these polygonal arcs strings.
Suppose that we divide the cube into two parts by an arbitrary plane E that is paralel 
to base of the cube B. Then, if E intersects each string at one and only one point, we 
say that these n strings in B are an n-braid.
Suppose that Bn is the set of all n-braids (to be more precise all the equivalence 
class of these braids). For two elements in Bn , i.e., for two n-braids a  and (3, it is 
possible to define a product for two n braids. First, glue the base of the cube that 
contains a  to the top face of the cube that contains p. The gluing together of the two
107 On The Braids for 810 Knot
Hakan Şimşek, Ahmet Küçük 108
In Garside genarators, i.e., positive braids, i.e., braids which are positive powers of 
the generators, Garside introduced the fiindemantal braid A:
A = (S, S2 • • • 5n_, ){SX S2 -  S„_2 )••■(<?, ¿2 )<V (l)
He showed that every element W  e  Bn can be represented algoritmically by a
word W of the form A ' P , where r is an integer and P  is a positive word, and r  is 
a maximal for all such representations (Birman, J., 1998).
Lemma 1: The word A# in S„ has the following properties:
(i) For each word V  £  S n , A #V  =  FA# .
(ii) For each j (1 < /  <  n -  1), D (A#) contains a word with initial letter sj 
and word with final letter Sj (Birman, J., 1974).
From now on the semigroup Sn will be identified with its image e(Sn ) in Bn .
Lemma 2: For each j ( (1 < j  <  n — 1) , there is a positive word X  j such that 
o v 1 =  . Also, A c t”1 = c r ~ l / A (which, together with Lemma 1, implies
that A  ]V  = VA  1 for every braid word V) (Birman, J., 1974).
Theorem 1 (Garside Solution to the Word Problem in Bn): If /?  e  Bn then (3 is 
represented by unique word of the form AmP  , where the integers m and the 
positive word P  are computed from any representative cr*1 • • • (7 ^  of the word p 
in the following manner:
(i) List the positive word X , , • • •, X n_, whose existence is established 
by Lemma 1.
(ii) Replace every letter <7~\i, which occurs in the braid word
^ ; - ^ i  b> A A’ u .
(iii) Using the property A 'V  = VA  '(Lemma 2) collect all A 1’s
introduced in (ii) at the left, so that (3 is represented by a word of the
form A k P0 , where P0 is positive. Note that k  < 0 .
(iv) Construct D(P0)
(v) In D(P0), choose a word A h P  such that h is maximal. Let m = h + k
(Notethat h > 0 )
(vi) Construct D(P). Let P  be base of D(P).
Proof: See (Birman, J., 1974).
109 On The Braids for 810 Knot
Now, let a  be a braid and let us connect, by a set o f parallel arcs that lie outside the 
square, the points A x, A 2, - - - , A n on the top of a rectengular diagram of a braid a
to the points A \  , A ' 2 , •1 •, A ' n respectively, on the bottom of the same diagram. 
Then in a natural way we form regular diagram of knot or link from a braid. A knot 
that has been created in this way is said to be a knot, K, created from the braid a.
Theorem 2 (Alexander’s Theorem): Given arbitrary (oriented) knot (or link), then 
it is equivalent (with orientation) to a knot (or link) that has been formed from a 
braid (Murasugi, K., 1996).
Proof: Let D be an oriented regular diagram of a knot K. Firstly cut the D at a point 
(not a crossing point) P(, , and then pull the loose ends apart so that we now have a 
(1,1) tangle. Figure 3.
We shall show that we can change this tangle into a braid a . The knot, in a sense 
induced as described previously from the braid, is equivalent to K. If the tangle T 
has m local maxima, then it also has m local minima. In the case m = 0, T is a 1- 
braid and so no proof is required.
Figure 3
So suppose that m > 0, then there exists an arc ab in T, which we may say “is rising 
upwards”, connecting a local minimum to a local maximum b. Figure 4.
Hakan Şimşek, Ahmet Küçük 110
Further we may assume that ab intersects with the other parts of the tangle at n
the arc a ja j+l intersects only one part of the tangle, see Figure 4(b). Next replace 
the arc by the much larger arc a 0P]'P[a ] . The large arc P\P{ lies outside
the tangle T, and the arcs a QP^ and a ]Pl are selected in such a way that if a 0a,
passes over (or under) the other segment, then they also pass over (or under) all the 
other segments. The result of the above manipulations is a (2,2) tangle. Figure 4(b).
It follows immediately that the oriented knot obtained by joining (outside the 
square) the four endpoints of this (2,2) tangle by curves is equivalent to original 
knot. By using the same methods as above, with regard to the arcs
does not have a local minimum and maximum at a and b respectively, and further 
has at most m  — 1 local maxima anda minima. Continuing this procedure, we shall 
finally create a tangle that has no local maxima and minima. This tangle is our
Definition: A knot is called positive, if it has a positive diagram, i.e., diagram with
111 On The Braids for 810 Knot
Lemma 3: If a positive n-string braid /?  involves a single occurence of some braid 
generators ô j (1 <  j < n - Y ) ,  then it has same closure as some positive
Lemma 4: If the closure of a positive n-braid is a knot or non-separated link, then 
its Conway polynomial is of degree c-n+1, where c is its number of braid crossing, n
Lemma 5: If K is a prime positive knot which has a Conway polynomial of degree 
2m, then a minimal string positive braid p which closes to K has at most 4 m  — 3
Now, we will find braids for 8 |0 knots. Regular diagram of 810 is given in Figure 5.
Figure 5
Then Theorem 2 is applied for the knot. Then we got 4-braids for this knot. 4 braid 
is given in Figure 6.
Hakan Şimşek, Ahmet Küçük 112
At the same time, we will write the Garside generators for this braid. We applied
Now we examine the positivity structure. Lemma 4 gives us that (3 braid's Conway 
is of degree 6. Also, Lemma 5 gives us its braids at most 9 crossings. But its braids
113 On The Braids for 810 Knot
REFERENCES
ARTIN, E. (1925).Theorie der Zopfe Hamburg Abh. 4, 47-72.
BIRMAN, J., K.H. KO and S.J. LEE (1998). “A New Aproach To The Word And 
Conjugacy Problems In The Braid Group”, Advances In Mathematics, vol. 139, 
(issue 2), 322-353.
BIRMAN, J. (1974). Braids,Links and Mapping Class Groups, Princeton Universty 
Press.
BUSKIRK, J.V. (1983). “Positive Knots Have Positive Conway Polynomials”,
Springer Lecture Notes in Mathematics, 1144, 146-159.
GARSIDE, F.A. (1969). “The Braid Group and Other Groups”, Quarterly Journal o f  
Mathematics, 20 No 78, 235-254.
MURASUGI., K. (1996). Knots Theory and Its Applications, Birkhauser: Boston.
PATERSON, M. S., and A.A. RASBOROV (1991). “The Set Minimal Braids is 
co_NP-Complete”, Journal o f Algorithms, 12, 393-408.


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