#16059: Equality vs hash for braid groups
-------------------------------------+-------------------------------------
Reporter: tmonteil | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.2
Component: group theory | Resolution:
Keywords: braid group, | Merged in:
hash, Cayley graph | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | 141adb09fe3e8f45079d706135dc071dc2e1eec7
u/tmonteil/equality_vs_hash_for_braid_groups| Stopgaps:
Dependencies: |
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Changes (by tmonteil):
* status: new => needs_review
* commit: => 141adb09fe3e8f45079d706135dc071dc2e1eec7
Old description:
> Playing with braid groups for a demo today, i want to see its Cayley
> graph in a neighbourhood of the neutral element:
>
> {{{
>
> ball = set()
> ball.add(B.one())
> for length in range(1,4):
> for w in Words(alphabet=B.gens(), length=length):
> ball.add(prod(w))
> G = B.cayley_graph(elements=ball, generators=B.gens()
> G.plot()
>
> }}}
>
> Hmmm, it looks locally like the free group, as if there was no relation !
>
> `s0 * s2` is a different vertex as `s2 * s0`, while:
>
> {{{
> sage: s0 * s2 == s2 * s0
> True
> }}}
>
> Indeed:
>
> {{{
> sage: hash(s0 * s2)
> 954209079
> sage: hash(s2 * s0)
> 1883627875
> }}}
>
> There should be a canonical representation for elements in Braid groups.
> At least, two equal elements should have the same hash. Currently,
> `cayley_graph` answers something wrong !
New description:
Playing with braid groups for a demo today, i want to see its Cayley graph
in a neighbourhood of the neutral element:
{{{
ball = set()
ball.add(B.one())
for length in range(1,4):
for w in Words(alphabet=B.gens(), length=length):
ball.add(prod(w))
G = B.cayley_graph(elements=ball, generators=B.gens())
G.plot()
}}}
[[Image(Cayley_before.png)]]
Hmmm, it looks locally like the free group, as if there was no relation !
`s0 * s2` is a different vertex as `s2 * s0`, while:
{{{
sage: s0 * s2 == s2 * s0
True
}}}
Indeed:
{{{
sage: hash(s0 * s2)
954209079
sage: hash(s2 * s0)
1883627875
}}}
There should be a canonical representation for elements in Braid groups.
At least, two equal elements should have the same hash. Currently,
`cayley_graph` answers something wrong !
After the modification, the Cayley group looks like:
[[Image(Cayley_after.png)]]
--
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Ticket URL: <http://trac.sagemath.org/ticket/16059#comment:3>
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