March 8, 2005
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MA4F2 Braid Groups
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Vague denition of braid groups
Figure 1: A braid on 4 strings
This is up
1.1 This section is purposely vague. Later well do things more precisely.
1.2 Vague denition of braids. Fix n 1. A braid on n strings, or an
n-br
MA241 Combinatorics Marking Sheet 1
Deadline: Wednesday, 26 January 2005, 2:00.
For this sheet, B1(b), B2, B6(abc) and B7 will be assessed.
Marks are in the margin. (Each assessment has 25 points in total.)
Correct answers always get full marks even if th
MA4F2 Braid Groups Sheet 4
Deadline: Friday, 11 March 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Let G, H be a groups and let
H G H
(x, g) xg
den
MA241 Combinatorics Marking Sheet 2
Deadline: Wednesday, 9 February 2005, 2:00.
For this sheet, (B2)cdef, (B4), (B5), (B6) will be assessed.
+
(B2) Let B3 be the monoid presented by (S1 , R1 ) = (1, 2 | 121 = 212).
(Warning: is the identity, 1 is not.) Le
THE UNIVERSITY OF WARWICK
FOURTH YEAR EXAMINATION: never
MOCK EXAM MA4F20 BRAID GROUPS 20042005
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are entered on each answer book.
Calculato
MA4F2 Braid Groups Sheet 3
Deadline: Wednesday, 2 March 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Let x = si and y = sj .
(a) Suppose |i j| = 1.
MA4F2 Braid Groups Sheet 1
Deadline: Wednesday, 26 January 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
(A1) Prove that the following spaces are path-co
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MA4F2 Braid Groups
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(c) Deduce a contradiction.
(d) Finish the proof.
6.10 Rewriting systems. Let (S, R) be a monoid presentation. The biinvariant closure of R is
:= (axb, ayb) (x, y) R, a, b S .
R
We call (S, R) a rewriting system if is
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MA4F2 Braid Groups
59
Figure 15:
j
i
(a): aij
(b):
16.13 Exercise. Suppose j < k < < i. We made a relation (16.12) as a
consequence of the relations (16.4) and (16.5) and whose left hand side starts
with aik and its right hand side with aj
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MA4F2 Braid Groups
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(gy)(gx)(gy). In a similar way, prove yourself that (gsi )(gsj ) = (gsj )(gsi ) if
|i j| > 1. This proves that v is well-dened.
It is clear that uv = 1, so that in particular v is injective. It remains
to show that v is
MA4F2 Braid Groups Sheet 2
Deadline: Wednesday, 9 February 2005, 2:00.
Solutions to Section B are for handing in. Please put your solutions into the
MA4F2 Braid Groups box in front of the Undergraduate Oce.
Needless to say, you may use previous parts of q
MA241 Combinatorics Marking Sheet 3
Deadline: Wednesday, 2 March 2005, 2:00.
For this sheet (B1), (B2), (B5), (B7)(cdef) are marked.
(B1) Let S denote the set of fundamental reections in 7 and write si < sj
if and only if i < j. Find the lexicographically