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 deniti
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 t
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
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
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 particul
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 O
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 Undergraduat
March 8, 2005
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)
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MA4F2 Braid Groups
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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 han
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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
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 Undergraduat
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 s