HDR

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HOW TO REDUCE PROBLEMS ON BRAIDS ? (Serge
Burckel) http//www2.univ-reunion.fr/burckel
Reduce order problems from Bn to Bn
No linear order on B3 can be compatible with
both products (left and right). PROOF
212
12 1
212
121
212
212
contradiction
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THEOREM (Dehornoy, Laver, B.) There exists a
unique linear order lt on B compatible with left
product such that for i ? n A positive braid on
?i1 , ?i2 , .... , ?n B in Bn A.B lt ?i
. ?i1 .... ?n (That implies ?1 lt ?2 lt ?3
...) It is a well-ordering of type It is a
restriction of DEHORNOYs order
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Combinatoric of this well-ordering
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Well-ordering on trees and words
lt
lt
3121 lt 3212 lt 1321
The well-ordering on B corresponds to this
order on minimal trees in the class of a braid.
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It is NOT CRP
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Is it the unique linear order CLP on B such
that ?1 lt ?2 lt ?3 ... ?
Is there also unicity on the braid groups ?
Partial answers.......
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THEOREM There exists a unique linear order
lt on braid groups compatible with left product
and such that for i ? n A positive braid on
?i1 , ?i2 , .... , ?n B in Bn AB lt ?i .
?i1 .... ?n (That implies ?1 lt ?2 lt ?3
...) It is the DEHORNOYs order.
i) It is a well-ordering... (FALSE ?1-1 gt ?1-2
gt ?1-3 gt .....) ii) In every infinite sequence t1
gt t2 gt t3 ... the minimal numbers of ? -1 in
the representations of the tk is not bounded.
Moreover, one of the followings holds
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PROOF reduce to positive braids. RECALL. For
every braid u u ? ?-n(u) . U U is a
positive braid, n(u)the minimal number of ? -1
in the possible representations of u.
One can compare two braids u,v in the
well-ordering on B u lt v ? ?-n(u) . U lt
?-n(v) . V For Mmax(n(u),n(v)) u lt v ? ?M .
?-n(u) . U lt ?M . ?-n(v) . V ? P  lt Q  (now
positive)
For an infinite sequence t1 gt t2 gt t3 ... if
max(n(tk)) lt ? then one can construct P1 gt P2 gt
P3 ........(positive) contradiction with the
well-ordering (B,gt)
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NEW WELL-ORDERINGS IN OTHER RESTRICTIONS 1.
(BK,lt) is a well-ordering. where BK is the
set of braids (with an arbitrary number of
strands) that admit a representation of length at
most K. 2. (BK-,lt) is a well-ordering. where
BK- is the set of braids (with an arbitrary
number of strands) that admit a representation
with at most K letters ? -1 . (B B0-)
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Linear Time Algorithm for the word problem on B3
1212121 2112111
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Rewriting Systems for B3