Braid Representatives: Difference between revisions

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{{Startup Note}}
{{Startup Note}}

{{HelpAndAbout1|n=2|s=BR}}
BR stands for Braid Representative. BR[k,l] represents a braid on k strands with crossings l={i1,i2,...}, where a positive index i within the list l indicates a right-handed crossing between strand number i and strand number i+1 and a negative i indicates a left handed crossing between strands numbers |i| and |i|+1. Each ij can also be a list of non-adjacent (i.e., commuting) indices. BR also acts as a "type caster": BR[K] will return a braid whose closure is K if K is given in any format that KnotTheory` understands. BR[K] where K is is a named knot with up to 10 crossings returns a minimum braid representative for that knot.
{{HelpAndAbout2|n=3|s=BR}}
The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See his article on the subject at arXiv:math.GT/0401051. Vogel's algorithm was implemented by Dan Carney in the summer of 2005 at the University of Toronto.
{{HelpAndAbout3}}


<!--$$?BR$$-->
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Revision as of 11:05, 24 August 2005

Every knot and every link is the closure of a braid. KnotTheory` can also represent knots and links as braid closures:

(For In[1] see Setup)

In[2]:= ?BR

BR stands for Braid Representative. BR[k,l] represents a braid on k strands with crossings l={i1,i2,...}, where a positive index i within the list l indicates a right-handed crossing between strand number i and strand number i+1 and a negative i indicates a left handed crossing between strands numbers |i| and |i|+1. Each ij can also be a list of non-adjacent (i.e., commuting) indices. BR also acts as a "type caster": BR[K] will return a braid whose closure is K if K is given in any format that KnotTheory` understands. BR[K] where K is is a named knot with up to 10 crossings returns a minimum braid representative for that knot.

In[3]:= BR::about

The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See his article on the subject at arXiv:math.GT/0401051. Vogel's algorithm was implemented by Dan Carney in the summer of 2005 at the University of Toronto.