Braid Representatives
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 s=BR]:= {{{s}}}::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. |
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In[2]:= 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]:= 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. |
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In[2]:= 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]:= 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. |
