Simple braids
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Simple braids Rehana Ashraf1 · Barbu Berceanu2 In memoriam S¸ tefan Papadima Received: 25 February 2019 / Revised: 4 January 2020 / Accepted: 10 March 2020 © Springer Nature Switzerland AG 2020
Abstract We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes. Keywords Positive braids · Square free braids · Conjugation classes of simple braids · Unlink Mathematics Subject Classification 20F36 · 57M25 · 57M27 · 05A05
1 Introduction and statement of results Artin braid group Bn [2], the geometrical analogue of the symmetric group n , is a central object of study, connected with various mathematical domains. See [8,12–14], and also [15] for a recent survey. Garside found a new solution of the word problem and solved the conjugacy problem in Bn , using the braid monoid MBn of positive braids [11]: this is generated by the positive braids xi , i = 1, . . . , n − 1,
This research was partially supported by Higher Education Commission of Pakistan.
B
Barbu Berceanu [email protected] Rehana Ashraf [email protected]
1
Lahore College for Women University, Jail Road, Lahore, Pakistan
2
Simion Stoilow Institute of Mathematics, P.O. Box 1-764, 014700 Bucharest, Romania
123
R. Ashraf, B. Berceanu 1
xi
i− 1
···
i
i+ 1
i+ 2
n
···
and has Artin defining relations xi x j = x j xi if |i − j| = 1 and xi+1 xi xi+1 = xi xi+1 xi . The Garside braid n = x1 (x2 x1 ) . . . (xn−1 xn−2 . . . x2 x1 ) plays a central role: for instance, n xi −1 n = x n−i , and the next four sets of positive braids coincide: the set of divisors of n (α | n ), Div(n ) = {α ∈ MBn | there exist δ, ε ∈ MBn , n = δαε}, the set of left divisors of n , (α |L n ), DivL (n ) = {α ∈ MBn | there exists ε ∈ MBn , n = αε}, the set of right divisors of n (α |R n ) DivR (n ) = {α ∈ MBn | there exists δ ∈ MBn , n = δα}, and also the set SFn of the square free elements in MBn (α ∈ MBn is square free if there is no generator xi such that xi2 | α). Also conjugation of positive braids in Bn is equivalent to conjugation in MBn (if α, β ∈ MBn and γ ∈ Bn are such that αγ = γβ, then there is δ ∈ MBn such that αδ = δβ) and this can be reduced to a sequence of conjugations with δ in Div(n ), see [8,11]. For positive braids α, β we write α | β or, equivalently, α ∈ Div(β), if there exist δ, ε ∈ MBn such that β = δαε. Let σ : MFn−1 → MBn be the natural projection (MFn−1 is the free monoid generated by x1 , x2 , . . . , xn−1 ). The diagram of β ∈ MBn , Dg(β), is the set of words in the free monoid MFn−1 representing β, that is σ −1 (β), see [8,11]. Computing polynomial invariants (Alexander-Conway, Jones, and also D) of closed braids, we found Fibonacci type recurrences which reduce computations to a new class of square free positive braids, see [3,7]. First we define five sets of positive braids: the set LSBn of literally simple braids, t
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