Solving the recognition problem of Lorenz braids via matrices of inversions for permutations
E. A. Elrifai and Redha. A. Alghamdi
Princess Nourah Bint Abdul Rahman University,
Kingdom of Saudi Arabia:
1- Department of Mathematical Sciences, College of Science.
2- Deanship of ScientiÖc Research. eaelrifai@pnu.edu.sa rdh.ashour24@hotmail.com
November 2016
Abstract
In this work, we present some needed results about matrices of inversions for permutations. Then we apply it for solving the recognition problem of Lorenz braids. Each Lorenz braid is uniquely determined by a unique simple binary matrix. Then, we got a quick algorithm for counting the trip number (minimal braid index) hence, crossing number and minimal braid representative of the
Lorenz…show more content… A positive braid is called a positive permutation braid if each pair of its strands cross in a positive sense at most once. The braid where each two strands cross each other exactly once in a positive sense is called the fundamental braid, denoted n:
Where,
1 = I; 2 = 1; 3 = 1:21; n = n1n1n2:::21;
2 n = (12:::n1) n Any oriented knot or link K can be viewed as a closed braid b, for a braid word in some Bn: The braid index for a knot or link is the smallest integer n, such that it can be represented as a closed nbraid.
DeÖnition 1 A lorenz braid L(l; r) is a Önite set of strands that embeds on the the
Lorenz braid template, Ögure 1c. It is a two groups of strands, a left group of l strands and a right group of r strands, l + r = n. It is a positive permutation braid with the restrictions, Strands in the same group never cross one another.
Strands in the left group always pass over those in the right group.
Each strand in any group should cross some strands in the other group.
The permutation at the end of strands and as a product of disjoint cycles
= 12
:::k
; k 2; no two cycles i = j of the same length s, and i
(x) =
j
(x) + t, x = 1; 2; :::; s, for some integer s. The permutation at the ends of…show more content… The recognition problem in mathematics is to decide whether an element belongs to a speciÖc category. Here our recognition problem is to decide whether a matrix of inversion for a permutation is a Lorenz matrix. In another words, can we recover
Lorenz braids hence Lorenz knots and links from their matrices of inversion for permutations. Theorem 3 A matrix M = (mij ) of inversions for a permutation = (12:::n) in Sn is a Lorenz matrix if and only if there exist two integers l; r; 1 < l; r < n; l+r = n, such that mij =
1 8 1 i l; l + 1 j l + i i 0 otherwise
Proof. Let be a Lorenz permutation, then we have two groups of strands with no crossing in each group. But i < j 8 1 i < j l and 8 l + 1 i < j n; then in both cases there are no inversions, which means that mij = 0 8 1 i < j l
(the l l submatrix at the Örst l rows and l columns of the matrix ML(l;r)
). Also mij = 0 8 l + 1 i < j n (the r r submatrix at the last r rows and r columns of the matrix ML(l;r)
). Also 1 < k 8 k > 1; k = l + 1; l + 2; :::; l + 1
1, then m1k =
1 8 k = l+1; l+2; :::; l+11:
In general i < k 8 k > i; k = l+1; l+2; :::;