Drawing Braids

2009-05-22T11:34:36Z

VidarElbor:

http://www.textaloloacelrel.com
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{{Startup Note}}



{{HelpLine|
n = 1 |
in = <nowiki>BraidPlot</nowiki> |
out= <nowiki>BraidPlot[br, opts] produces a plot of the braid br. Possible options
are Mode, HTMLOpts, WikiOpts and Images.</nowiki>}}


Thus for example,



{{In|
n = 2 |
in = <nowiki>br = BR[5, {{1,3}, {-2,-4}, {1, 3}}];</nowiki>}}




{{Graphics|
n = 4 |
in = <nowiki>Show[BraidPlot[br]]</nowiki> |
img= Drawing_Braids_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}


<code>BraidPlot</code> takes several options:



{{InOut|
n = 5 |
in = <nowiki>Options[BraidPlot]</nowiki> |
out= <nowiki>{Mode -> Graphics, Images -> {0.gif, 1.gif, 2.gif, 3.gif, 4.gif},

HTMLOpts -> , WikiOpts -> }</nowiki>}}


The <code>Mode</code> option to <code>BraidPlot</code> defaults to "<code>Graphics</code>", which produces output as above. An alternative is setting <code>Mode -> "HTML"</code>, which produces an HTML <tt><table></tt> that can be readily inserted into HTML documents:



{{InOut|
n = 6 |
in = <nowiki>BraidPlot[br, Mode -> "HTML"]</nowiki> |
out= <nowiki><table cellspacing=0 cellpadding=0 border=0>
<tr><td><img src=1.gif><img src=0.gif><img src=1.gif></td></tr>
<tr><td><img src=2.gif><img src=3.gif><img src=2.gif></td></tr>
<tr><td><img src=1.gif><img src=4.gif><img src=1.gif></td></tr>
<tr><td><img src=2.gif><img src=3.gif><img src=2.gif></td></tr>
<tr><td><img src=0.gif><img src=4.gif><img src=0.gif></td></tr>
</table></nowiki>}}


The table produced contains an array of image inclusions that together draws the braid using 5 fundamental building blocks: a horizontal "unbraided" line (<tt>0.gif</tt> above), the upper and lower halves of an overcrossing (<tt>1.gif</tt> and <tt>2.gif</tt> above) and the upper and lower halves of an underfcrossing (<tt>3.gif</tt> and <tt>4.gif</tt> above).

Assuming <tt>0.gif</tt> through <tt>4.gif</tt> are [[Image:BraidPart0.gif]], [[Image:BraidPart1.gif]], [[Image:BraidPart2.gif]], [[Image:BraidPart3.gif]] and [[Image:BraidPart4.gif]], the above table is rendered as follows:

{| cellspacing=0 cellpadding=0 border=0 align=center
|[[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]]
|-
|[[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]]
|-
|[[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]
|-
|[[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]]
|-
|[[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]]
|}

The meaning of the <code>Images</code> option to <code>BraidPlot</code> should be clear from reading its default definition:



{{InOut|
n = 7 |
in = <nowiki>Images /. Options[BraidPlot]</nowiki> |
out= <nowiki>{0.gif, 1.gif, 2.gif, 3.gif, 4.gif}</nowiki>}}


The <code>HTMLOpts</code> option to <code>BraidPlot</code> allows to insert options within the HTML <tt><img></tt> tags. Thus



{{InOut|
n = 8 |
in = <nowiki>BraidPlot[BR[2, {1, 1}], Mode -> "HTML", HTMLOpts -> "border=1"]</nowiki> |
out= <nowiki><table cellspacing=0 cellpadding=0 border=0>
<tr><td><img border=1 src=1.gif><img border=1 src=1.gif></td></tr>
<tr><td><img border=1 src=2.gif><img border=1 src=2.gif></td></tr>
</table></nowiki>}}


The above table is rendered as follows:

<center>[[Image:BraidPlotHTMLExample2.gif]]</center>



{{HelpLine|
n = 9 |
in = <nowiki>CollapseBraid</nowiki> |
out= <nowiki>CollapseBraid[br] groups together commuting generators in the braid
br. Useful in conjunction with BraidPlot to produce compact braid plots.</nowiki>}}


Thus compare the plots of <code>br1</code> and <code>br2</code> below:



{{InOut|
n = 10 |
in = <nowiki>br1 = BR[TorusKnot[5, 4]]</nowiki> |
out= <nowiki>BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]</nowiki>}}




{{Graphics|
n = 12 |
in = <nowiki>Show[BraidPlot[br1]]</nowiki> |
img= Drawing_Braids_Out_11.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 13 |
in = <nowiki>br2 = CollapseBraid[BR[TorusKnot[5, 4]]]</nowiki> |
out= <nowiki>BR[4, {{1}, {2}, {3, 1}, {2}, {3, 1}, {2}, {3, 1}, {2}, {3, 1}, {2},

{3}}]</nowiki>}}




{{Graphics|
n = 15 |
in = <nowiki>Show[BraidPlot[br2]]</nowiki> |
img= Drawing_Braids_Out_14.gif |
out= <nowiki>-Graphics-</nowiki>}}

Knot Atlas - User contributions [en]

Drawing Braids