Abstract: In this study we introduce the retraction and conditional retraction of braids and braid groups, we show the retraction of braid group is not necessary a braid group also a retraction of a singular braid is not necessary a singular braid. We prove that a retraction of a braid is a braid and every retraction of a braid group is a monoid also we prove that a retraction is a braid invariant. The limit of all types of retraction is described.
INTRODUCTION
A braid theory introduced by E. Artin at 1925 and the concept of braid theory was found to have applications in other fields after the 1950s and this gave fresh impetus to the study of braids. More studies on braid theory are studied by many researches by this braid theory has gradually been prospected, refined and polished. In mathematics it is, now, recognized from one of the basic theories and is of benefit in such branches as topology and algebraic geometry. Also, it is of profound use in other areas of the sciences physics, statistical mechanics, chemistry and biology. The braid group was took an important role in this field. The iridescent hue of this concepts flowering into full bloom and activity occurred in 1984, when V. Jones put into action with inordinate success the original aim of Artin. i.e. the application of braids to knot theory. In (El-Ghoul et al., 2006, 2007) we introduced a new direction on knot theory called folding and retraction of knot. More studies of retraction in El-Ghoul (1985, 1995, 1998, 2002). In this study our intention introduce the concepts of retraction on the braid theory and braid group, continuations to the two articles above, we study the effect of retraction and conditional retraction on braid and braid group and singular braid.
DEFINITIONS
Here we will show some definitions and basic concepts which we will use it in the main results.
Definition 1
Let D be a unit cube, so D = {(x, y, z): 0 ≤ x, y, z ≤ 1} on the top face of cube place n points, a1, a2, ..., an and similarly, place n points on the bottom face b1, b2, ..., bn Now,join the points a1, a2, ..., an with b1, b2, ..., bn by means of n arcs d1, d2, ..., dn (as smooth curves), this arcs are mutually disjoint and each d1 connects some aj to bk (J ≠K or J = K) not connect aj to aK or bj to bK. Each plane Es, such that z = s, 0 ≤ s ≤ 1 (parallel to xy-plane intersections each arc di at one and only one point.
A configuration of n arcs d1, d2, ..., dn
with end points a1, a2, ..., an and b1,
b2, ..., bn is called n-braid or a braid with n
strings Fig. 1 (Murasugi, 1996).
| Fig. 1: | Representation of braid in unit cube |
| Fig. 2: | Product of β1 and β2 |
Definition 2
Let Bn be a set of all n-braids and β1, β2 ε Bn, we may create a third n-braid from them which we shall call their product and denoted by β1β2 as follows:
Glue the bottom arcs of β1 with the top arcs of β2 (Murasugi and Kurpita, 1999) Fig. 2.
Remark 1
| • | For β1 ε Bn there is β1-1 such that β1β1-1 = e ε Bn Fig. 3. |
| • | β1β2 ≠β2β1 |
| • | The product of braids is associative, i.e., (β1β2)β3 ≈1(β2β2) |
Theorem 1
The set of all n-braids, Bn forms a group. This group is usually called the n-braid group or Artin's n-braid group (Murasugi and Kurpita, 1999; Gemein, 2001).
Theorem 2
For any n ≤ 1 the n-braid group Bn has the following presentation:
Where, σi denotes the standard generator of the braid
group (Kauffman, 1991; Murasugi and Kurpita, 1999; Gemein, 2001) Fig.
4.
| Fig. 3: | Product of β1 and β1-1 and β1 ° β2 ≠β2 ° β1 |
| Fig. 4: | Standard generator σi |
Definition 3
An n-dimensional manifold is a Housdorff topological space M, such that every point of M has a neighborhood homeomorphic to open set U ⊂ Rn (Munkers, 1975).
Definition 4
Let A be a subset of a topological space X. A continuous map r: X →A is said to be retraction if r(a) = a for all a ε A (Massay, 1967; Munkers, 1975).
Definition 5
A subset A ⊂ X is deformations retract of X if there is a retraction r: X →A such that i ° r homotopic to the identity map. That is, there exists a continuous function f: X x [0, 1] →X such that for x ε X, f (x, 0) = x and f (x, 1) = r (x) and for all a ε A and all t ε [0, 1], f (a, t) = a (Massay, 1967).
Definition 6
Let β1, β2 be two n-braids in a cube D, we say β1 ambient isotopic to β2, denoted by β1 ≈ β2, if there exists a homeomorphism H: Dx[0,1]→Dx [0,1], such that H(x,t) = ht((x),t), tε[0,1], ht: D→D and h0(β1) = β1, ht(β2) = β2 (Murasugi and Kurpita, 1999).
Definition 7
Let β be a n-braid and suppose the ith string di of β joins ai to bj(i) for i = 1, 2, 3, ….n. Define g:
Bn →Sn (the set af all permutations of the set {1,
2, 3, 4, ….., n} as,
then β is called a pure n-braid (Murasugi and Kurpita, 1999).
Remark 2
If β1 ≈ β2 ⇒ π(β1) = π(β2).
Theorem 3
Let Bn be a n-braid group and Sn the symmetry
group of n elements, then there exist a natural surjective homomorphism
f from Bn onto Sn, takes any braid β to the
permutation determined by β, f (β) = π (β).
The kernel of f is a pure n-braid group, denoted by Pn and Bn/Pn is isomorphic to Sn, the index of Pn in Bn is a finite, [Bn:Pn] = n! (Murasugi and Kurpita, 1999).
THE MAIN RESULTS
Let β = σi be a n-braid and r a retraction from β-{a} onto β', where a is a point on one arc of β, then we have two cases:
| • | β' is a trivial n-1-braid, if a ε di or a ε di+1. |
| • | β' = σi as a n-1-braid if a ε dj, j ≠i, i + 1 Fig. 5. |
If the retraction r from β-{a1, a2, ..., an} onto β' by remove all points from top level or bottom level or together, then we have two cases:
| • | β' is a trivial n-braid. |
| • | β' = σi as a n-braid Fig. 6. |
Theorem 4
A retract of any braid by remove a point or (points) from any arc
or (arcs) is a braid.
Proof
Let β be a n-braid, β = {d1, d2, ..., dn}
where di is a string and r be a continuous map from β-{a}
onto β' where a is a point on di, i = 1, 2, ..., n defined by
Theorem 5
Let β = σ1σ2.....σk
be a n-braid and r: (β-a) →β' be a retraction then β'
takes three cases:
| • | Trivial n-1-braid, if a ε d1. |
| • | β' = σ1σ2....σk-1 as a n-1-braid, if a εdi, 1 < i ≤ k + 1. |
| • | β' = σ1σ2....σk as a n-1-braid, if a ε di, i > k + 1. |
| Fig. 5: | Two cases of retraction of σi by remove one point |
| Fig. 6: | Two cases of retraction of σi by remove more then one point |
Proof
| • | Since d' 1 of σ1 glue with d1'' of σ2, also d1'' glue with d1''' of σ3…., d1k-1 of σk-1 glue with d1k of σk and since d1 = d1'd1''d1'''....d1k-1d1k hence if we remove d1, then all σ1, σ2, ..., σk are finished. |
| • | Since di, 1 < i ≤ k + 1 is not glue with any arc of any σ, then the only σ which finished is σi-1 and σi takes its place and represents σi in the retract. |
| • | Since all di, i > k + 1 are straight strings then remove it not finish any σ, hence the retract β' = σ1σ2...σk. |
Corollary 1
The limit of retractions of i) in Theorem 5 is a 1-braid group, also
the limit of the retraction of ii) is a 1-braid, but the limit of the
retractions of iii) is σ1σ2....σk.
Lemma 1
Let β, β1, β2 be a three elements of a
n-braid group Bn and r be a retraction from Bn-{a} onto
| • | r(β1 ° β2) = r(β1) ° r(β2), |
| • | r(β-1) = (r(β))-1, β-1 be the inverse element of β. |
We can show the lemma by the following examples:
| • | Let β1 = σ3σ1
and β2 = σ1σ2σ3σ1
be a two elements of 4-braid group B4, r be a retraction from
B4-{a} onto |
| • | Let β = σ2σ1σ3
be an element of 4-braid group B4 and β-1 = σ3-1σ1-1σ2-1
be its inverse, r be a retraction from B4-{a} onto |
| Fig. 7: | r(β1 ° β2) ≠r(β1) ° r(β2) |
Theorem 6
The retraction of a n-braid group Bn is not necessary a
braid group.
Proof
From the lemma above we can say easy there is not exist for all element
in r(Bn-{a}) an inverse element.
Theorem 7
Let Bn be a n-braid group and M = {σ1, σ2,
..., σn-1} be a set of its generators, r be a retraction from
Bn-{a} onto
Proof
| • | If we remove an arc d1 from all the generators, then σ1 was vanished, also if we remove dn, then σn-1 was vanished and σ2 becomes σ1, etc. |
| • | Proof ii) came directory from the proof i). |
Theorem 8
Let β = σ2σ4σ6...σ2k-2
be a n-braid, then β' = σ1σ3...σ2k-1
as a n-1-braid is a retract of β.
Proof
Let β = σ2σ4σ6...σ2k-2
be a n-braid, {d1, d2, ..., dn} be an arcs
of β and r be a continuous map from β-{a} onto β' where a is
a point on di.i =1, 2, ..., n defined, by
Theorem 9
Let β = σ1σ3...σ2k-1
be a n-braid, then β = σ2σ4σ6...σ2k-2
as a n-1-braid is a retract of β.
| Fig. 8: | r(β-1) ≠(r(β))-1 |
Proof
The proof is similar to the proof of Theorem 8.
Theorem 10
Let Bn be a pure n-braid group generated by a1, a2,
..., an-1, where, ai = (σn-1σn-2,
..., σi+1)
| • | bi is a trivial n-1-braid if a ε di or a ε dn. |
| • | if a ε dj, i < j < n. |
| • | bi = ai-1 in n-1-braid, if a ε dj, j < i |
Proof
The proof is clear.
Theorem 11
Let Bn be a n-braid group and r be a retraction from Bn-{a}
onto
Proof
Let Bn be a n-braid group and r be a retraction from Bn-{a}
onto
Theorem 12
A retract of a singular braid not necessary a singular braid.
Proof
We can show that by the following Example:
Example 1
Let β = σ1σ3σ2 be a singular
4-braid and r: (β-{a}) →
| Fig. 9: | Singular braid not necessary a singular braid |
| Fig. 10: | r(β1) = r(β2) but β1
|
| Fig. 11: | β1, β2, β3 are equivalents and has the same r(βi-{a}), i = 1, 2, 3 |
Theorem 13
A retraction of any isotopic braids is an invariant.
Proof
Let β1, β2 be a two n-braids such that β1
≈ β2 and a Γ be a set of all n-braids and r be a
retraction from Γ-{a} onto
Remark 1
If r(β1) = r(β2), then not necessary β1 ≈ β2.
By the following example we show that:
Example 2
Let
Example 3
Let β1, β2, β3 be a three 4-braids,
easy we can get one from the another by some elementary moves, hence this three
4-braids are equivalents, if we apply the retraction on the three 4-braids,
r:(βi-{a}) →