#15843: Add reduced Burau representation, Alexander polynomial to Braid
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Reporter: | Type: enhancement
lipshitz | Priority: minor
Status: new | Component: algebra
Milestone: | Keywords: Braid, braid group, Burau
sage-6.2 | representation, Alexander polynomial, knot
Resolution: | theory
Report Upstream: N/A |
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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.)
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Ticket URL: <http://trac.sagemath.org/ticket/15843>
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