Positive braids and Lorenz links
In this work a new foundation for the study of positive braids in
Artin's braid groups is given. The basic braids considered are the
set SBn of positive permutation braids, defined as those positive
braids where each pair of arcs cross at most once. These are shown
to be in 1-1 correspondence with the permutations in S . A canonical
form for positive braids as products of braids in SB is given, ton
gether with an explicit algorithm for writing every positive braid in
canonical form and a practical test for use in the algorithm. This is
a useful approach to braid theory because permutations can be particularly
Applications of this canonical form are:
(1) An easily handled approach to Garside's solution of the word
problem in B .
(2) An algorithm to decide whether (/1 ) k is a factor of a positive
braid; this happens if and only if at most k canonical factors have
equal to /1 n (where /1 n is the positive braid with each pair of arcs
cross exactly once).
(3) A proof that a positive braid is a factor of (/1 ) k if and only if
its canonical form has at most k factors.
(4) An improvement of Garside's solution of the conjugacy problem,
this is by reducing the summit set to a much smaller invariant
class under conjugation (super summit set). This includes a necessary
and sufficient condition for positive braid to contain /1
up to conjugation.
The linear generators of the Hecke algebras used by Morton. and/
Short are in 1-1 correspondence with the elements of SB. The
canonical forms above give a quick proof that the number of strands
in a twist positive braid (one of the form (/1 )2mp for positive braid
P and for positive integer m) is the braid index of the closure of that
braid, which was first proved by Franks and Williams. It is also
shown that if the 2-variable link invariant P
(v, z) for an oriented link
L has width k in the variable v, then it is the same as the polynomial
of a closed k-braid, for k = 1, 2. A complete list of 3-braids of width
2, which close to knots, is given. It is also shown that twist positive
3-braids do not admit exchange moves (in the sense of Birman).
Consequently the conjugacy class of a twist positive 3-braid representative
is a complete link invariant, provided that Birman's conjecture
about Markov's moves and exchange moves holds.
Lorenz knots and links are studied as an example of positive links.
It is proved that a positive permutation braid 1T is a Lorenz braid if
and only if all braid words which equal 1T have the same single starting
letter. A semicanonical form for a minimal braid representative of a Lorenz link is given. It is proved that every algebraic link of c
components is a Lorenz link, for c = 1, 2. (The case for knots was
first proved by Birman and Williams). Consequently a necessary and
sufficient condition for a knot to be algebraic is given, together with
a canonical form for a minimal braid representative for every algebraic
knot. To some extent the relation between Lorenz knots and their
companions is studied.
It is shown that Lorenz knots and links of braid index 3 are determined
by conjugacy classes in B 3. A complete list of 3 -braids which
close to Lorenz knots and links is given and a complete list of pure
4-braids which close to Lorenz links is also given. It is shown that
Lorenz knots and links of braid index 3 are determined by their
Alexander polynomials. As a further analogy with the properties of
algebraic links it is shown that the linking pattern of a Lorenz link
L with pure braid representative and braid index t ~4, determines a
unique braid representative for L and so determines L.