#20760: Mirror image of links
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Reporter: mmarco | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.3
Component: algebraic | Resolution:
topology | Merged in:
Keywords: days74 | Reviewers: Travis Scrimshaw
Authors: Miguel Marco | Work issues:
Report Upstream: N/A | Commit:
Branch: | bf99b0819473d08585de76be80949d3404008fc8
public/knots/mirror_image-20760 | Stopgaps:
Dependencies: |
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Comment (by mmarco):
I am not sure it is a good idea. If the original link was created with a
pd code or gauss code, the computed braid might need more crossings than
the original link. If we compute the new link from the braid, we might be
unnecesarily increasing the number of crossings. Consider, for instance
the monster unknot:
{{{
sage: L = Link([[3,1,2,4], [8,9,1,7], [5,6,7,3], [4,18,6,5],
....: [17,19,8,18], [9,10,11,14], [10,12,13,11],
....: [12,19,15,13], [20,16,14,15], [16,20,17,2]])
sage: L
Link with 1 component represented by 10 crossings
sage: L.braid()
s0*s1^-1*s2^-1*s3^-1*s2*s1^-1*s0^-1*s1*s2^2*s1^-1*s3*s2*s1^-3
sage: len(L.braid().Tietze())
16
}}}
If we construct the mirror image from the braid, we sill loose the
posibility of computing some invariants using the simpler representation
with 10 crossings (instead of 16).
If we use the pd_code to compute the mirror image, we are safe on that
size: the resulting link will have exactly the same internal
representation as the original one.
--
Ticket URL: <http://trac.sagemath.org/ticket/20760#comment:4>
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