#15843: Add reduced Burau representation, Alexander polynomial to Braid
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Reporter: lipshitz | Owner:
Type: enhancement | Status: new
Priority: minor | Milestone: sage-6.2
Component: algebra | Resolution:
Keywords: Braid, braid | Merged in:
group, Burau representation, | Reviewers:
Alexander polynomial, knot theory | Work issues:
Authors: | Commit:
Report Upstream: N/A | bec075f7b708d1856750eebe835b5d2cd1d0d33b
Branch: | Stopgaps:
u/lipshitz/ticket/15843 |
Dependencies: |
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Changes (by lipshitz):
* commit: => bec075f7b708d1856750eebe835b5d2cd1d0d33b
Old description:
> Instances of Braid class could already return the image of the braid
> under the Burau representation. This patch adds code to optionally
> return the image under the reduced Burau representation. Using the
> reduced Burau representation, it can also return the (unnormalized)
> Alexander polynomial of the braid closure of the Braid.
>
> Examples of the new functionality from the doc_strings:
>
> sage: B = BraidGroup(4)
> sage: B.inject_variables()
> Defining s0, s1, s2
> sage: s0.burau_matrix(reduced=True)
> [-t 0 0]
> [-t 1 0]
> [-t 0 1]
> sage: b = B([1,1,1,3,3,2,-3,-1,-1,2,-1,-3,-2])
> sage: b.alexander_polynomial() #The Kinoshita-Terasaka knot.
> -1/t
>
> (The Kinoshita-Terasaka knot has Alexander polynomial 1; the
> unnormalized Alexander polynomial is only well-defined up to
> multiplication by plus or minus t^n.)
New description:
Instances of Braid class could already return the image of the braid under
the Burau representation. This patch adds code to optionally return the
image under the reduced Burau representation. Using the reduced Burau
representation, it can also return the (unnormalized) Alexander polynomial
of the braid closure of the Braid.
Examples of the new functionality from the doc_strings:
sage: B = BraidGroup (4) [[BR]] sage: B.inject_variables() [[BR]]
Defining s0, s1, s2 [[BR]] sage: s0.burau_matrix(reduced=True)[[BR]]
[-t 0 0][[BR]] [-t 1 0][[BR]] [-t 0 1] [[BR]] sage: b =
B([1,1,1,3,3,2,-3,-1,-1,2,-1,-3,-2]) [[BR]] sage:
b.alexander_polynomial() #The Kinoshita-Terasaka knot. [[BR]] -1/t
(The Kinoshita-Terasaka knot has Alexander polynomial 1; the unnormalized
Alexander polynomial is only well-defined up to multiplication by plus or
minus t to the n.)
--
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Ticket URL: <http://trac.sagemath.org/ticket/15843#comment:2>
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