ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 12 Jan 2019 08:47:54 -0600braid closureshttp://ask.sagemath.org/question/45011/braid-closures/ Hi! I've noticed that when taking the closure of a braid on $n$ strands, the resulting link is obtained by ignoring any "unused" strand:
B = BraidGroup(3)
print B([1]).components_in_closure(), Link(B([1])).number_of_components()
outputs 2 1, instead of 2 2.
In the documentation for the braid group I've read "The behavior of removing unused strands from an element of a braid group may change without notice in the future. Do not rely on this feature."
Is there a workaround that allows to have the same number of components in a braid and in its closure?
Note that braids given as an array of generators (as in the example above) are automatically simplified, so closing the braid [2,-2] (which should give the $2$-component unlink) produces the unknot. (In particular, the empty braid on $n$ strands should close to the $n$-component unlink.)
Fri, 11 Jan 2019 06:45:51 -0600http://ask.sagemath.org/question/45011/braid-closures/Answer by mmarco for <p>Hi! I've noticed that when taking the closure of a braid on $n$ strands, the resulting link is obtained by ignoring any "unused" strand:</p>
<pre><code>B = BraidGroup(3)
print B([1]).components_in_closure(), Link(B([1])).number_of_components()
</code></pre>
<p>outputs 2 1, instead of 2 2.
In the documentation for the braid group I've read "The behavior of removing unused strands from an element of a braid group may change without notice in the future. Do not rely on this feature."</p>
<p>Is there a workaround that allows to have the same number of components in a braid and in its closure?
Note that braids given as an array of generators (as in the example above) are automatically simplified, so closing the braid [2,-2] (which should give the $2$-component unlink) produces the unknot. (In particular, the empty braid on $n$ strands should close to the $n$-component unlink.)</p>
http://ask.sagemath.org/question/45011/braid-closures/?answer=45021#post-id-45021You can conjugate the braid so there are no unused strands anynore. In order to prevent the simplification, you can use two expressions of the same braid.
In your example:
cb1 = B([1,2,1])
cb2 = B([-2,-1,-2])
mybraid = B([1])
myconjbraid = cb1*mybraid*cb2
myconjbraid produces the same link as mybraid, but since it has no unused strands, it will contain also the trivial components.
Sat, 12 Jan 2019 06:20:31 -0600http://ask.sagemath.org/question/45011/braid-closures/?answer=45021#post-id-45021Comment by danieleC for <p>You can conjugate the braid so there are no unused strands anynore. In order to prevent the simplification, you can use two expressions of the same braid.</p>
<p>In your example:</p>
<p>cb1 = B([1,2,1])
cb2 = B([-2,-1,-2])</p>
<p>mybraid = B([1])</p>
<p>myconjbraid = cb1<em>mybraid</em>cb2</p>
<p>myconjbraid produces the same link as mybraid, but since it has no unused strands, it will contain also the trivial components.</p>
http://ask.sagemath.org/question/45011/braid-closures/?comment=45023#post-id-45023Thanks, that works perfectly!Sat, 12 Jan 2019 08:47:54 -0600http://ask.sagemath.org/question/45011/braid-closures/?comment=45023#post-id-45023